  In a previous work, the second-named author gave a complete description of
the action of automorphisms on the ordinary irreducible characters of the
finite symplectic groups. We generalise this in two directions. Firstly, using
work of the first-named author, we give a complete description of the action of
Galois automorphisms on irreducible characters. Secondly, we extend both
descriptions to cover the case of special orthogonal groups. As a consequence,
we obtain explicit descriptions for the character fields of symplectic and
special orthogonal groups.
