Let \(G\) be a a finite group, $p$ a prime, and $P$ a Sylow $p$-subgroup of $G$. A recent refinement, due to G. Navarro, of the McKay conjecture suggests that there should exist a bijection between irreducible characters of $p'$-degree of $G$ and irreducible characters of $p'$-degree of $N_G(P)$ which commutes with certain Galois automorphisms. This Galois automorphism refinement of the McKay conjecture has several interesting consequences. I will discuss my progress on proving one of these consequences, namely a way to read off from the character table of $G$ whether a Sylow 2-subgroup of $G$ is self-normalizing.