The mapping class group Mod(S) of a surface S acts on the homology H_1(S), yielding the well-studied symplectic representation Mod(S) → Sp(2g,Z). In this talk, we discuss an equivariant refinement of the symplectic representation. Namely, given a finite group G acting on S, the symplectic representation restricts to a map from the centralizer of G in Mod(S) to the centralizer of G in Sp(2g,Z). The image of this restriction has been studied by many authors and is generally difficult to understand. We discuss our result that the image of this restriction is arithmetic when S has genus at most 3.