On the mapping class group action on the homology of surface covers

Let $\phi \in \mathrm{M}\mathrm{o}\mathrm{d}(\mathrm{\Sigma })$ be an arbitrary element of the mapping class group of a closed orientable surface $\mathrm{\Sigma }$ of genus at least $2$. For any characteristic cover $\tilde{\mathrm{\Sigma }}\to \mathrm{\Sigma }$ one can consider the linear subspace ${\mathrm{H}}_1^{f\mathrm{.}o\mathrm{.}}(\tilde{\mathrm{\Sigma }},\mathbb{Q}{)}^{\phi }\subseteq {\mathrm{H}}_1(\tilde{\mathrm{\Sigma }},\mathbb{Q})$ consisting of all homology classes with finite $\phi$-orbit. We prove that $\dim {\mathrm{H}}_1^{f\mathrm{.}o\mathrm{.}}(\tilde{\mathrm{\Sigma }},\mathbb{Q}{)}^{\phi }$ can be arbitrary large for any fixed $\phi \in \mathrm{M}\mathrm{o}\mathrm{d}(\mathrm{\Sigma })$.