Let $p: S\to S'$ be a finite $G$-cover between closed oriented surfaces with branch locus $B$. Let $H_1^\mathrm{scc}(S;Q)$ the $G$-submodule
of the homology group $H_1(S;Q)$ generated by cycles supported on lifts of simple closed curves on $S\setminus B$. In a joint work with Andy Putman and
Nick Salter, we show that $H_1^\mathrm{scc}(S;Q)$ is a symplectic subspace of $H_1(S;Q)$ and that the centralizer of the group $G$
in the mapping class group of $S$ acts on the nonzero elements of $H_1^\mathrm{scc}(S;Q)$ with infinite orbits. Though this is not true
in general, thanks to these results, one can prove that $H_1^\mathrm{scc}(S;Q)= H_1(S;Q)$ in many interesting cases.