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

Abstract

Let $A$ and $G$ be finite groups. Suppose that $A$ acts coprimely on $G$ stabilizing $N\triangleleft G$. Let $\theta \in \rm{Irr}(N)$ be $A$-invariant. We prove that the number of $A$-invariant irreducible characters of $G$ that lie over $\theta$ can be counted in terms of the $(A, \theta)$-good conjugacy classes of $G_\theta/N$, where $G_\theta$ is the inertia subgroup of $\theta$ in $G$. This result generalizes a classic result of Gallagher and can be used to prove the following: if $P$ is an $A$-invariant Sylow $p$-subgroup of $G$ and $G$ is $p$-solvable, then there exists an $A$-equivariant (McKay) bijection between the irreducible characters of degree prime to $p$ of $G$ and those of $\textbf{N}_G(P)$. While this is a consequence of a recent result of D. Rossi, our approach here is independent of 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.