On some rational extension properties for $G{L}_n(q)$ and even-degree characters fixed by order-2 Galois automorphisms

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 $G{L}_n(q)$, $q$ odd, to the group extended by the transpose-inverse automorphism and prove that unipotent characters of $PS{L}_n(q)$ extend to rational characters of its automorphism group.