  Let $S$ be a closed oriented surface and $G$ a finite group of orientation
preserving automorphisms of $S$ whose orbit space has genus at least $2$. There
is a natural group homomorphism from the $G$-centralizer in $Diff^+(S)$ to the
$G$-centralizer in $Sp(H_1(S))$. We give a sufficient condition for its image
to be a subgroup of finite index and a weaker condition for this to have no
finite nonzero orbit (the Putman-Wieland property).
