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][1].  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.


  [1]: http://arxiv.org/PS_cache/math/pdf/0111/0111139v1.pdf