  We verify the inductive McKay condition for simple groups of Lie type C,
namely finite projective symplectic groups. This contributes to the program of
a complete proof of the McKay conjecture for all finite groups via the
reduction theorem of Isaacs-Malle-Navarro and the classification of finite
simple groups. In an important step we use a new counting argument to determine
the stabilizers of irreducible characters of a finite symplectic group in its
outer automorphism group. This is completed by analogous results on characters
of normalizers of Sylow d-tori in those groups.
