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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.