It appears that similar question has been asked at sci.math.research Tue, 19 Oct 1999. The answer by G. Kuperberg is quite interesting. Hope no one don't mind if I put it here:

As Torsten Ekedahl explained, it is sometimes the wrong question, but in modified form, the answer is sometimes yes.

For example, consider A_5, or its central extension Gamma = SL(2,5). The two 3-dimensional representations are Galois conjugates and there is no way to choose one or the other in association with the conjugacy classes. However, if you choose an embedding pi of Gamma in SU(2), then there is a specific bijection given by the McKay correspondence. The irreducible representations form an extended E_8 graph where two representations are connected by an edge if you can get from one to the other by tensoring with pi. The conjugacy classes also form and E_8 graph if you resolve the singularity of the algebraic surface C^2/Gamma. The resolution consists of 8 projective lines intersecting in an E_8 graph. If you take the unit 3-sphere S^3 in C^2, then the resolution gives you a surgery presentation of the 3-manifold S^3/Gamma. The surgery presentation then gives you a presentation of Gamma itself called the Wirtinger presentation. As it happens, each of the Wirtinger generators lies in a different non-trivial conjugacy class. In this way both conjugacy classes and irreps. are in bijection with the vertices of E_8.