I'm not sure this answers your question, but here is a general procedure:

Fix any representation $W$ of $G\times H$. Then for any representation $V$ of $H$, the hom space $\hom_H(W,V)$ carries a $G$-action, and hence produces a functor $H\mathrm{-Rep}\to G\mathrm{-Rep}$ (here, note that I do not require $H\subset G$). Some examples:

• Suppose $H\subset G$, and let $W=\mathbb C[G]$, which carries a $G\times H$-action, by making $G$ act by left multiplication and $H$ act by right multiplication. Then $\hom_H(\mathbb C[G],V)$ is exactly the induced representation $V^G$.
• Suppose $H\supset G$, and let $W=\mathbb C[H]$, which carries a $G\times H$-action, as above. Then $\hom_H(\mathbb C[H],V)=V$ is simply the restriction of the $H$-action to $G$.
• A bit different in flavor, but Deligne-Lusztig theory provides a similar construction. Letting $X_n$ be the variety $(\det(t_i^{q^{j-1}})_{i,j})^{q-1}=(-1)^{n-1}$, the étale cohomology $H_c^i(X_n,\overline{\mathbb Q}_\ell)$ carries an action of $\mathrm{GL}_n(\mathbb F_q)\times\mathbb F_{q^n}^\times$, which allows the construction of a representation of $\mathrm{GL}_n(\mathbb F_q)$ from a character of $\mathbb F_{q^n}^\times$.

I'm not sure this answers your question, but here is a general procedure:

Fix any representation $W$ of $G\times H$. Then for any representation $V$ of $H$, the hom space $\hom_H(W,V)$ carries a $G$-action, and hence produces a functor $H\mathrm{-Rep}\to G\mathrm{-Rep}$ (here, note that I do not require $H\subset G$). Some examples:

• Suppose $H\subset G$, and let $W=\mathbb C[G]$, which carries a $G\times H$-action, by making $G$ act by left multiplication and $H$ act by right multiplication. Then $\hom_H(\mathbb C[G],V)$ is exactly the induced representation $V^G$.
• Suppose you have a homomorphism $\varphi\colon G\to H$, and let $W=\mathbb C[H]$, which carries a $G\times H$-action, as above (explicitly, $(g,h)\cdot x:=\varphi(g)xh^{-1}$). Then the $G$-action on $\hom_H(\mathbb C[H],V)=V$ is the composition $G\xrightarrow{\varphi}H\xrightarrow\pi\mathrm{GL}(V)$, where $\pi$ is the representation of $H$. In particular: if $G\subseteq H$, then this is the restriction, and if $G\to H=G/N$ is a quotient, then this is the inflation.
 

• A bit different in flavor, but Deligne-Lusztig theory provides a similar construction. Letting $X_n$ be the variety $(\det(t_i^{q^{j-1}})_{i,j})^{q-1}=(-1)^{n-1}$, the étale cohomology $H_c^i(X_n,\overline{\mathbb Q}_\ell)$ carries an action of $\mathrm{GL}_n(\mathbb F_q)\times\mathbb F_{q^n}^\times$, which allows the construction of a representation of $\mathrm{GL}_n(\mathbb F_q)$ from a character of $\mathbb F_{q^n}^\times$.

I'm not sure this answers your question, but here is a general procedure:

Fix any representation $W$ of $G\times H$. Then for any representation $V$ of $H$, the hom space $\hom_H(W,V)$ carries a $G$-action, and hence produces a functor $H\mathrm{-Rep}\to G\mathrm{-Rep}$ (here, note that I do not require $H\subset G$). Some examples:

• Suppose $H\subset G$, and let $W=\mathbb C[G]$, which carries a $G\times H$-action, by making $G$ act by right multiplication and $H$ act by right multiplication. Then $\hom_H(\mathbb C[G],V)$ is exactly the induced representation $V^G$.
• Suppose $H\supset G$, and let $W=\mathbb C[H]$, which carries a $G\times H$-action, as above. Then $\hom_H(\mathbb C[H],V)=V$ is simply the restriction of the $H$-action to $G$.
• A bit different in flavor, but Deligne-Lusztig theory provides a similar construction. Letting $X_n$ be the variety $(\det(t_i^{q^{j-1}})_{i,j})^{q-1}=(-1)^{n-1}$, the étale cohomology $H_c^i(X_n,\overline{\mathbb Q}_\ell)$ carries an action of $\mathrm{GL}_n(\mathbb F_q)\times\mathbb F_{q^n}^\times$, which allows the construction of a representation of $\mathrm{GL}_n(\mathbb F_q)$ from a character of $\mathbb F_{q^n}^\times$.

I'm not sure this answers your question, but here is a general procedure:

Fix any representation $W$ of $G\times H$. Then for any representation $V$ of $H$, the hom space $\hom_H(W,V)$ carries a $G$-action, and hence produces a functor $H\mathrm{-Rep}\to G\mathrm{-Rep}$ (here, note that I do not require $H\subset G$). Some examples:

• Suppose $H\subset G$, and let $W=\mathbb C[G]$, which carries a $G\times H$-action, by making $G$ act by left multiplication and $H$ act by right multiplication. Then $\hom_H(\mathbb C[G],V)$ is exactly the induced representation $V^G$.
• Suppose $H\supset G$, and let $W=\mathbb C[H]$, which carries a $G\times H$-action, as above. Then $\hom_H(\mathbb C[H],V)=V$ is simply the restriction of the $H$-action to $G$.
• A bit different in flavor, but Deligne-Lusztig theory provides a similar construction. Letting $X_n$ be the variety $(\det(t_i^{q^{j-1}})_{i,j})^{q-1}=(-1)^{n-1}$, the étale cohomology $H_c^i(X_n,\overline{\mathbb Q}_\ell)$ carries an action of $\mathrm{GL}_n(\mathbb F_q)\times\mathbb F_{q^n}^\times$, which allows the construction of a representation of $\mathrm{GL}_n(\mathbb F_q)$ from a character of $\mathbb F_{q^n}^\times$.