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There is in fact such a connection: for every $\mathbb{Z}G$-bimodule $M$ there exists a $G$-module $U(M)$ such that the Hochschild cohomology of $\mathbb{Z}G$ with coefficients in $M$ is naturally isomorphic to the group cohomology of $G$ with coefficients in $U(M)$. This group cohomology can also be considered as the cohomology of the groupoid ${\bf B}G$ (with one object whose automorphism group is $G$) with coefficients in the local system of abelian groups determined by $U(M)$.

To see this relation observe that there is an adjunction between $G$-modules and $\mathbb{Z}G$-bimodules defined as follows: the right adjoint $$U: \mathbb{Z}G-{\rm BiMod} \to G-{\rm Mod}$$ sends a $\mathbb{Z}G$-bimodule $M$ to the $G$-module $M$ with the action given by $(g,m) \mapsto gmg^{-1}$, and the left adjoint $L: G-{\rm Mod} \to \mathbb{Z}G-{\rm BiMod}$ sends a $G$-module $M$ to the $\mathbb{Z}G$-bimodule $$L(M) := M \otimes \mathbb{Z}G = \oplus_{g \in G}M\left<g\right>$$ where the right action is given by $(\sum_g m_g\left<g\right>,h) \mapsto \sum_g m_g\left<gh\right>$ and the left action is given by $(h,\sum_g m_g\left<g\right>) \mapsto \sum_gh(m_g)\left<hg\right>$. It then follows from general categorical considerations that for every $\mathbb{Z}G$-bimodule $M$ there is a natural isomorphism $$ {\rm HH}^n(\mathbb{Z}G,M) = {\rm Ext}_{\mathbb{Z} G-{\rm BiMod}}(\mathbb{Z}G,M) = {\rm Ext}_{\mathbb{Z} G-{\rm BiMod}}(L(\mathbb{Z}),M) \cong $$ $$ {\rm Ext}_{G-{\rm Mod}}(\mathbb{Z},U(M)) = {\rm H}^n(G,U(M)). $$ In particular, the Hochschild cohomology ${\rm HH}^n(\mathbb{Z}G) := {\rm HH}^n(\mathbb{Z}G,\mathbb{Z}G)$ is naturally isomorphic to the group cohomology of $G$ with coefficients in the $G$-module $\mathbb{Z}G$ equipped with the conjugation action.

There is also a conceptual take on this connection: it comes from interpreting the group cohomology of $G$ as the Quillen cohomology of the ($\infty$-)-groupoid ${\bf B}G$ and the Hochschild cohomology of $\mathbb{Z}G$ as the Quillen cohomology the dg-category ${\bf B}(\mathbb{Z}G)$. The connection then arises simply from the natural adjunction between $\infty$-groupoids and dg-categories.

For every $\mathbb{Z}G$-bimodule $M$ there exists a $G$-module $U(M)$ such that the Hochschild cohomology of $\mathbb{Z}G$ with coefficients in $M$ is naturally isomorphic to the group cohomology of $G$ with coefficients in $U(M)$.

To see this observe that there is an adjunction between $G$-modules and $\mathbb{Z}G$-bimodules defined as follows: the right adjoint $$U: \mathbb{Z}G-{\rm BiMod} \to G-{\rm Mod}$$ sends a $\mathbb{Z}G$-bimodule $M$ to the $G$-module $M$ with the action given by $(g,m) \mapsto gmg^{-1}$, and the left adjoint $L: G-{\rm Mod} \to \mathbb{Z}G-{\rm BiMod}$ sends a $G$-module $M$ to the $\mathbb{Z}G$-bimodule $$L(M) := M \otimes \mathbb{Z}G = \oplus_{g \in G}M\left<g\right>$$ where the right action is given by $(\sum_g m_g\left<g\right>,h) \mapsto \sum_g m_g\left<gh\right>$ and the left action is given by $(h,\sum_g m_g\left<g\right>) \mapsto \sum_gh(m_g)\left<hg\right>$. It then follows from general categorical considerations that for every $\mathbb{Z}G$-bimodule $M$ there is a natural isomorphism $$ {\rm HH}^n(\mathbb{Z}G,M) = {\rm Ext}_{\mathbb{Z} G-{\rm BiMod}}(\mathbb{Z}G,M) = {\rm Ext}_{\mathbb{Z} G-{\rm BiMod}}(L(\mathbb{Z}),M) \cong $$ $$ {\rm Ext}_{G-{\rm Mod}}(\mathbb{Z},U(M)) = {\rm H}^n(G,U(M)). $$ In particular, the Hochschild cohomology ${\rm HH}^n(\mathbb{Z}G) := {\rm HH}^n(\mathbb{Z}G,\mathbb{Z}G)$ is naturally isomorphic to the group cohomology of $G$ with coefficients in the $G$-module $\mathbb{Z}G$ equipped with the conjugation action.

When $G$ is finite the group cohomology $H^n(G, \mathbb{Z}G)$ is isomorphic to the cohomology $H^n(aG,\mathbb{Z})$ of the adjoint groupoid $aG$, and so the connection that you state holds in higher dimensions. I'm not sure what happens when $G$ is infinite.

Remark (of a somewhat unrelated nature):

The connection above comes from interpreting the group cohomology of $G$ as the Quillen cohomology of the ($\infty$-)-groupoid ${\bf B}G$ and the Hochschild cohomology of $\mathbb{Z}G$ as the Quillen cohomology the dg-category ${\bf B}(\mathbb{Z}G)$. The connection then arises from the natural adjunction between $\infty$-groupoids and dg-categories.

There is in fact such a connection: for every $\mathbb{Z}G$-bimodule $M$ there exists a $G$-module $U(M)$ such that the Hochschild cohomology of $\mathbb{Z}G$ with coefficients in $M$ is naturally isomorphic to the group cohomology of $G$ with coefficients in $U(M)$. This group cohomology can also be considered as the cohomology of the groupoid ${\bf B}G$ (with one object whose automorphism group is $G$) with coefficients in the local system of abelian groups determined by $U(M)$.

To see this relation observe that there is an adjunction between $G$-modules and $\mathbb{Z}G$-bimodules defined as follows: the right adjoint $$U: \mathbb{Z}G-{\rm BiMod} \to G-{\rm Mod}$$ sends a $\mathbb{Z}G$-bimodule $M$ to the $G$-module $M$ with the action given by $(g,m) \mapsto gmg^{-1}$, and the left adjoint $L: G-{\rm Mod} \to \mathbb{Z}G-{\rm BiMod}$ sends a $G$-module $M$ to the $\mathbb{Z}G$-bimodule $$L(M) := M \otimes \mathbb{Z}G = \oplus_{g \in G}M\left<g\right>$$ where the right action is given by $(\sum_g m_g\left<g\right>,h) \mapsto \sum_g m_g\left<gh\right>$ and the left action is given by $(h,\sum_g m_g\left<g\right>) \mapsto \sum_gh(m_g)\left<hg\right>$. It then follows from general categorical considerations that for every $\mathbb{Z}G$-bimodule $M$ there is a natural isomorphism $$ {\rm HH}^n(\mathbb{Z}G,M) = {\rm Ext}_{\mathbb{Z} G-{\rm BiMod}}(\mathbb{Z}G,M) = {\rm Ext}_{\mathbb{Z} G-{\rm BiMod}}(L(\mathbb{Z}),M) \cong $$ $$ {\rm Ext}_{G-{\rm Mod}}(\mathbb{Z},U(M)) = {\rm H}^n(G,U(M)). $$ In particular, the Hochschild cohomology ${\rm HH}^n(\mathbb{Z}G) := {\rm HH}^n(\mathbb{Z}G,\mathbb{Z}G)$ is naturally isomorphic to the group cohomology of $G$ with coefficients in the $G$-module $\mathbb{Z}G$ equipped with the conjugation action.

There is also a conceptual take on this adjunction: it is the adjunction between the Quillen cohomology of the groupoid ${\bf B}G$ and the Quillen cohomology the ${\rm Ab}$-enriched category ${\bf B}(\mathbb{Z}G)$ associated to the adjunction between groupoids and ${\rm Ab}$-enriched categories.

There is in fact such a connection: for every $\mathbb{Z}G$-bimodule $M$ there exists a $G$-module $U(M)$ such that the Hochschild cohomology of $\mathbb{Z}G$ with coefficients in $M$ is naturally isomorphic to the group cohomology of $G$ with coefficients in $U(M)$. This group cohomology can also be considered as the cohomology of the groupoid ${\bf B}G$ (with one object whose automorphism group is $G$) with coefficients in the local system of abelian groups determined by $U(M)$.

To see this relation observe that there is an adjunction between $G$-modules and $\mathbb{Z}G$-bimodules defined as follows: the right adjoint $$U: \mathbb{Z}G-{\rm BiMod} \to G-{\rm Mod}$$ sends a $\mathbb{Z}G$-bimodule $M$ to the $G$-module $M$ with the action given by $(g,m) \mapsto gmg^{-1}$, and the left adjoint $L: G-{\rm Mod} \to \mathbb{Z}G-{\rm BiMod}$ sends a $G$-module $M$ to the $\mathbb{Z}G$-bimodule $$L(M) := M \otimes \mathbb{Z}G = \oplus_{g \in G}M\left<g\right>$$ where the right action is given by $(\sum_g m_g\left<g\right>,h) \mapsto \sum_g m_g\left<gh\right>$ and the left action is given by $(h,\sum_g m_g\left<g\right>) \mapsto \sum_gh(m_g)\left<hg\right>$. It then follows from general categorical considerations that for every $\mathbb{Z}G$-bimodule $M$ there is a natural isomorphism $$ {\rm HH}^n(\mathbb{Z}G,M) = {\rm Ext}_{\mathbb{Z} G-{\rm BiMod}}(\mathbb{Z}G,M) = {\rm Ext}_{\mathbb{Z} G-{\rm BiMod}}(L(\mathbb{Z}),M) \cong $$ $$ {\rm Ext}_{G-{\rm Mod}}(\mathbb{Z},U(M)) = {\rm H}^n(G,U(M)). $$ In particular, the Hochschild cohomology ${\rm HH}^n(\mathbb{Z}G) := {\rm HH}^n(\mathbb{Z}G,\mathbb{Z}G)$ is naturally isomorphic to the group cohomology of $G$ with coefficients in the $G$-module $\mathbb{Z}G$ equipped with the conjugation action.

There is also a conceptual take on this connection: it comes from interpreting the group cohomology of $G$ as the Quillen cohomology of the ($\infty$-)-groupoid ${\bf B}G$ and the Hochschild cohomology of $\mathbb{Z}G$ as the Quillen cohomology the dg-category ${\bf B}(\mathbb{Z}G)$. The connection then arises simply from the natural adjunction between $\infty$-groupoids and dg-categories.

Yes. There is an adjunction between $G$-modules and $\mathbb{Z}G$-bimodules defined as follows: the right adjoint $U: \mathbb{Z}G-{\rm BiMod} \to G-{\rm Mod}$ sends a $\mathbb{Z}G$-bimodule $M$ to the $G$-module $M$ with the action given by $(g,m) \mapsto gmg^{-1}$, and the left adjoint $L: G-{\rm Mod} \to \mathbb{Z}G-{\rm BiMod}$ sends a $G$-module $M$ to the $\mathbb{Z}G$-bimodule $L(M) := M \otimes \mathbb{Z}G = \oplus_{g \in G}M\left<g\right>$ where the right action is given by $(\sum_g m_g\left<g\right>,h) \mapsto \sum_g m_g\left<gh\right>$ and the left action is given by $(h,\sum_g m_g\left<g\right>) \mapsto \sum_gh(m_g)\left<hg\right>$. It then follows from general categorical considerations that for every $\mathbb{Z}G$-bimodule $M$ there is a natural isomorphism $$ {\rm HH}^n(\mathbb{Z}G,M) = {\rm Ext}_{\mathbb{Z} G-{\rm BiMod}}(\mathbb{Z}G,M) = {\rm Ext}_{\mathbb{Z} G-{\rm BiMod}}(L(\mathbb{Z}),M) \cong $$ $$ {\rm Ext}_{G-{\rm Mod}}(\mathbb{Z},U(M)) = {\rm H}^n(G,U(M)). $$ In particular, the Hochschild cohomology ${\rm HH}^n(\mathbb{Z}G) := {\rm HH}^n(\mathbb{Z}G,\mathbb{Z}G)$ is naturally isomorphic to the group cohomology of $G$ with coefficients in the $G$-module $\mathbb{Z}G$ equipped with the conjugation action. This group cohomology can also be considered as the cohomology of the space (or groupoid) $BG$ with coefficients in the local system of abelian groups $\mathbb{Z}G$.

There is in fact such a connection: for every $\mathbb{Z}G$-bimodule $M$ there exists a $G$-module $U(M)$ such that the Hochschild cohomology of $\mathbb{Z}G$ with coefficients in $M$ is naturally isomorphic to the group cohomology of $G$ with coefficients in $U(M)$. This group cohomology can also be considered as the cohomology of the groupoid ${\bf B}G$ (with one object whose automorphism group is $G$) with coefficients in the local system of abelian groups determined by $U(M)$.

To see this relation observe that there is an adjunction between $G$-modules and $\mathbb{Z}G$-bimodules defined as follows: the right adjoint $$U: \mathbb{Z}G-{\rm BiMod} \to G-{\rm Mod}$$ sends a $\mathbb{Z}G$-bimodule $M$ to the $G$-module $M$ with the action given by $(g,m) \mapsto gmg^{-1}$, and the left adjoint $L: G-{\rm Mod} \to \mathbb{Z}G-{\rm BiMod}$ sends a $G$-module $M$ to the $\mathbb{Z}G$-bimodule $$L(M) := M \otimes \mathbb{Z}G = \oplus_{g \in G}M\left<g\right>$$ where the right action is given by $(\sum_g m_g\left<g\right>,h) \mapsto \sum_g m_g\left<gh\right>$ and the left action is given by $(h,\sum_g m_g\left<g\right>) \mapsto \sum_gh(m_g)\left<hg\right>$. It then follows from general categorical considerations that for every $\mathbb{Z}G$-bimodule $M$ there is a natural isomorphism $$ {\rm HH}^n(\mathbb{Z}G,M) = {\rm Ext}_{\mathbb{Z} G-{\rm BiMod}}(\mathbb{Z}G,M) = {\rm Ext}_{\mathbb{Z} G-{\rm BiMod}}(L(\mathbb{Z}),M) \cong $$ $$ {\rm Ext}_{G-{\rm Mod}}(\mathbb{Z},U(M)) = {\rm H}^n(G,U(M)). $$ In particular, the Hochschild cohomology ${\rm HH}^n(\mathbb{Z}G) := {\rm HH}^n(\mathbb{Z}G,\mathbb{Z}G)$ is naturally isomorphic to the group cohomology of $G$ with coefficients in the $G$-module $\mathbb{Z}G$ equipped with the conjugation action.

There is also a conceptual take on this adjunction: it is the adjunction between the Quillen cohomology of the groupoid ${\bf B}G$ and the Quillen cohomology the ${\rm Ab}$-enriched category ${\bf B}(\mathbb{Z}G)$ associated to the adjunction between groupoids and ${\rm Ab}$-enriched categories.