The McKay conjecture and Brauer's induction theorem

Let G be an arbitrary finite group. The McKay conjecture asserts that G and the normalizer NG (P) of a Sylow p-subgroup P in G have the same number of characters of degree not divisible by p (that is, of p′-degree). We propose a new refinement of the McKay conjecture, which suggests that one may choose a correspondence between the characters of p′-degree of G and NG (P) to be compatible with induction and restriction in a certain sense. This refinement implies, in particular, a conjecture of Isaacs and Navarro. We also state a corresponding refinement of the Broué abelian defect group conjecture. We verify the proposed conjectures in several special cases.