  In this note, we prove that if every character of a finite group $G$ fixed by
an order-2 Galois automorphism has odd degree, then $G$ has a normal Sylow
$2$-subgroup. On the way, we study extensions of characters of $GL_n(q)$, $q$
odd, to the group extended by the transpose-inverse automorphism and prove that
unipotent characters of $PSL_n(q)$ extend to rational characters of its
automorphism group.
