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 $\theta$ be a $P$-invariant irreducible character of $N$. Suppose that $G\mathrm{/}N$ 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 $\theta$ of degree prime to $p$ of $G$ and ${\mathbf{\text{N}}}_G(P)$. 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 $\theta$.