This question is pretty self contained in the title. Let $G$ be a group regarded as a category with a single object and consider a functor $F:G \to Cat$. This exactly the data of a $G$-category that I will denote by $\mathcal{C}$. This has an obvious notion of $H$-fixed points category, $\mathcal{C}^H$ for $H \leq G$.

I would like to know if given a category $\mathcal{A}$ one could find a $G$-category $\mathcal{C}$ such that $\mathcal{C}^H=Fun(H,\mathcal{A})$ for every subgroup $H \leq G$.