  Assume $G$ is a finite symplectic group $\mathrm{Sp}_{2n}(q)$ over a finite
field $\mathbb{F}_q$ of odd characteristic. We describe the action of the
automorphism group $\mathrm{Aut}(G)$ on the set $\mathrm{Irr}(G)$ of ordinary
irreducible characters of $G$. This description relies on the equivariance of
Deligne--Lusztig induction with respect to automorphisms. We state a version of
this equivariance which gives a precise way to compute the automorphism on the
corresponding Levi subgroup; this may be of independent interest. As an
application we prove that the global condition in Sp\"ath's criterion for the
inductive McKay condition holds for the irreducible characters of
$\mathrm{Sp}_{2n}(q)$.
