  Let $G$ be a finite symmetric, general linear, or general unitary group
defined over a field of characteristic coprime to $3$. We construct a canonical
correspondence between irreducible characters of degree coprime to $3$ of $G$
and those of $N_{G}(P)$, where $P$ is a Sylow $3$-subgroup of $G$. Since our
bijections commute with the action of the absolute Galois group over the
rationals, we conclude that fields of values of character correspondents are
the same.
