Chern approximations for generalised group cohomology

Let G be a finite group and E is a suitable generalised cohomology theory. We define and study a ring C(E,G) that is the best-possible approximation to E0BG that can be built using knowledge of the complex representations of G. We give a description of Q⊗C(E,G) in terms of generalised characters, and we study some cases in which the map C(E,G)→E0BG is an isomorphism.