The McKay conjecture with group automorphisms and the Okuyama-Wajima argument

Let N be normal subgroup of a finite group G , p be a prime, P be a Sylow p -subgroup of G and θ be a P -invariant irreducible character of N . Suppose that G/N is a p -solvable group. In this note we show that, whenever a finite group A acts on G stabilizing P , there exists an A -equivariant McKay bijection between irreducible characters lying over θ of degree prime to p of G and NG(P)N. This is a consequence of a recent result of D. Rossi. Our approach here is independent from Rossi's and follows the original idea of the proof of the McKay conjecture for p -solvable groups. In particular, we rely on the so-called Okuyama-Wajima argument to deal with characters above Glauberman correspondents. For this purpose, we generalize a classical result of P. X. Gallagher on the number of irreducible characters of G lying over θ .