Let $G$ be a finite group. Let $\mathcal{C}=Rep-G$ be the rigid $\mathbb{C}$-linear symmetric monoidal category of finite dimensional complex representations of $G$.
Can we recover some homological data about $G$ from $\mathcal{C}$? That is: can we recover for example the homology groups $H_n(G,\mathbb{Z})$ or maybe $H^n(G,\mathbb{C}^{\times})$ just from $\mathcal{C}$?
Of course that one can argue that, by Deligne's Theorem, one can reconstruct $G$ out of $\mathcal{C}$, and then ask what is the homology of $G$. I ask if there is something more direct.
One example is that the group of invertible object of $\mathcal{C}$ is isomorphic with the group $H^1(G,\mathbb{C}^{\times})$, but I do not know what happens beyond that.
