How often can a finite group be realized as a Galois group over a field?

Let G be any finite group and C any class of fields. By nu(C)(G) we denote the minimal number of realizations of G as a Galois group over some field from the class C. For G abelian and C the class of algebraic extensions of Q we give an explicit formula for nu(C) (G). Similarly we treat the case of an abelian p-group G and the class epsilon(p) which is conjectured to be the class of all fields of characteristic not equal p for which the Galois group of the maximal p-extension is finitely generated. For non-abelian groups G we offer a variety of sporadic results.