  Given a finite group $G$ acting on a surface $S$, the centralizer of G in the
mapping class group $\textrm{Mod}(S)$ has a natural representation given by its
action on the homology $H_1(S; \mathbb{Q})$. We consider the question of
whether this representation has arithmetic image. Several authors have given
positive and negative answers to this question. We give a complete answer when
S has genus at most 3.
