Navarro has conjectured a necessary and sufficient condition for a finite group G to have a self-normalizing Sylow 2-subgroup, which is given in terms of the behavior of the ordinary irreducible characters of G under a specific Galois automorphism. Navarro-Tiep-Vallejo have conjectured a similar statement regarding groups whose Sylow 2-normalizers contain a single irreducible 2-Brauer character. Thanks to reduction theorems proved by myself and Navarro-Vallejo, respectively, a large part of the proofs of these conjectures is to understand the action of this Galois automorphism on characters of groups of Lie type. I will discuss the recent proof of these conjectures, including a description of the action of Galois automorphisms on characters of groups of Lie type