I am reading this article, where they classify the semisimple module categories with finitely many irreducibles over the category of representations of quantum $SL(2)$.
They say that, in the classical case, this is in analogy with finite subgroups of $SL(2)$. Indeed, when $H\subset G$ is a subgroup, the category $\mathrm{Rep}_k(H)$ becomes a module category over $\mathrm{Rep}_k(G)$, where $\mathrm{Rep}_k(G)\times\mathrm{Rep}_k(H)\to\mathrm{Rep}_k(H)$ is given by $(\rho,\mu)\mapsto\mathrm{Res}_H^G(\rho)\otimes\mu$.
Now my question is: is this really analogous to finite subgroups in the classical case? I.e., are the only semisimple module categories with finitely many irreducibles over $\mathrm{Rep}_k(G)$ given by $\mathrm{Rep}_k(H)$, where $H\subset G$ is a finite subgroup?
Additionally, is it possible to reconstruct this subgroup $H$ when we are just given a semisimple module category with finitely many irreducibles?