Navarro's refinement of the McKay conjecture

Let $G$ be a finite group, let $p$ be a prime, and let $P$ be a Sylow $p$-subgroup of $G$. Write $\operatorname{Irr}_{p'}(G)$ for the set of irreducible characters of $G$ whose degrees are not divisible by $p$, and similarly for $\operatorname{Irr}_{p'}(\operatorname{N}_G(P))$. For a nonnegative integer $e$, let $\sigma \in \operatorname{Gal}(\mathbb{Q}_{|G|}/\mathbb{Q})$ be an $(e,p)$-Galois automorphism.

Navarro's refinement. There exists a bijection between the $\sigma$-invariant characters in

$$\operatorname{Irr}_{p'}(\operatorname{N}_G(P))$$

and the $\sigma$-invariant characters in

$$\operatorname{Irr}_{p'}(G).$$

This refines the McKay conjecture by requiring compatibility with specified Galois automorphisms. The stated consequence was proved by Navarro, Tiep and Turull, so this formulation is resolved.