  We prove a result that relates the number of homomorphisms from the
fundamental group of a compact nonorientable surface to a finite group $G$,
where conjugacy classes of the boundary components of the surface must map to
prescribed conjugacy classes in $G$, to a sum over values of irreducible
characters of $G$ weighted by Frobenius-Schur multipliers. The proof is
structured so that the corresponding results for closed and possibly orientable
surfaces, as well as some generalizations, are derived using the same methods.
We then apply these results to the specific case of the symmetric group.
