The farthest I got thinking about this problem (and I haven't thought about it all that much) is that module categories over C[G] are classified in Section 3.4. They correspond to pairs K a subgroup of G and a choice of central extension of K (or equivalently, a certain cohomology class). In the case where there's no central extension, the dual category is some sort of Hecke algebra category C[K\G/K]-mod that I've never totally understood. Also I don't know how to modify that construction when you introduce the central extension. Anyway, modulo understanding those issues the question comes down to when a twisted Hecke algebra category C[K\G/K]-mod can be equivalent as a tensor category to C[H]-mod for some group H.

The farthest I got thinking about this problem (and I haven't thought about it all that much) is that module categories over $\mathbb C[G]$ are classified in Section 3.4 of Ostrik - Module categories, weak Hopf algebras, and modular invariants. They correspond to pairs $K$ a subgroup of $G$ and a choice of central extension of $K$ (or equivalently, a certain cohomology class). In the case where there's no central extension, the dual category is some sort of Hecke algebra category $\text{$\mathbb C[K\backslash G/K]$-mod}$ that I've never totally understood. Also I don't know how to modify that construction when you introduce the central extension. Anyway, modulo understanding those issues, the question comes down to when a twisted Hecke algebra category $\text{$\mathbb C[K\backslash G/K]$-mod}$ can be equivalent as a tensor category to $\text{$\mathbb C[H]$-mod}$ for some group $H$.