I haven't seen this question in standard textbooks, so I decide to give it a try here. It might relate to deeper structures of certain TQFTs, but I'm not sure.

Let $G$ be a finite group. Its finite-dimensional complex representations are well understood, the regular representation $\mathbb{C}[G]$ being the most important one. Note that it's not only a vector space acted by $G$: it also has a natural algebra structure and naturally compatible with $G$-action.

###Question
Is there a classification of group representations that have compatible algebra structures?