A refinement of the McKay conjecture

Let G be a finite group and N be the normalizer of a Sylow p-subgroup of G. The McKay conjecture, which has been open for more than 30 years, states that G and N have the same number of irreducible characters of degree not divisible by p (i.e. of p'-degree). The conjecture has been strengthened in a number of ways. In particular, a version due to Isaacs and Navarro suggests the existence of a correspondence between irreducible character degrees of G and of N modulo p and up to sign, if one considers only characters of p'-degree. I will review these conjectures and will discuss a possible new refinement, which implies the Isaacs—Navarro conjecture.