  G. 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 ordinary irreducible characters of $G$. In a previous article, the
author has reduced the proof of this conjecture to showing that certain related
statements hold for simple groups. In this article, we describe the action of
Galois automorphisms on the Howlett-Lehrer parametrization of Harish-Chandra
induced characters. We use this to complete the proof of the conjecture by
showing that the remaining simple groups satisfy the required conditions.
