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For a fusion category $\mathcal{C}$ the Brauer-Picard group $\text{BrPic}(\mathcal{C})$ is the group of all invertible $\mathcal{C}$-bimodule categories under multplication $\boxtimes_\mathcal{C}$.

Let the fusion category $\mathcal{C}$ be a $G$-extension of the fusion category $\mathcal{D}$, that is:

$\mathcal{C} = \bigoplus_{g \in G} \mathcal{C}_g$

$\mathcal{C}_e = \mathcal{D}$

It is shown in [Etingoff, Nikshych, & Ostrick, Fusion categories and homotopy theory (arXiv:0909.3140), Thm. 6.1] that $\mathcal{C}_g$ is an invertible $\mathcal{D}$-bimodule category for each $g \in G$ and that $\mathcal{C}_g \boxtimes_\mathcal{D} \mathcal{C}_h \cong \mathcal{C}_{gh}$ as $\mathcal{D}$-bimodule categories, hence defining a homomorphism:

$G \to \text{BrPic}(\mathcal{D})$

$g \mapsto \mathcal{C}_g$

(It is further proven that $G$-extensions of $\mathcal{D}$ are in one to one correspondence with morphisms of categorical 2-groups $G \to \underline{\underline{\text{BrPic}}}(\mathcal{D})$ in [ditto, Thm 7.7], where the target is the Brauer-Picard groupoid)


Question: Suppose the $G$-extension $\mathcal{C}$ is group theoretical, i.e. it is categorically Morita dual to a pointed fusion category. What (if anything) can be said about the resulting homomorphism $G \to \text{BrPic}(\mathcal{D})$? Or the resulting morphism of categorical 2-groups? What if we restrict $\mathcal{D}$ to be pointed?

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If $\mathcal{C}$ is group-theoretical, then $\mathcal{D}$ is group-theoretical. So suppose $\mathcal{D}$ is group-theoretical and denote by $X$ the set of equivalence classes of pointed $\mathcal{D}$-module categories. The group $BrPic(\mathcal{D})$ acts on $X$ using the tensor product of module categories over $\mathcal{D}$. For a $G$-extension with associated group morphism $\rho: G\to BrPic(\mathcal{D})$, the G-extension is group-theoretical if and only if there is a fix point ($X^G\neq \emptyset$). For details you can see Theorem 1.2.

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