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

Let φ ∈ Mod(Σ) be an arbitrary element of the mapping class group of a closed orientable surface Σ of genus at least $2$. For any characteristic cover Σ → Σ one can consider the linear subspace H₁f.o.(Σ, Q)^φ ⊆ H₁(Σ, Q) consisting of all homology classes with finite φ-orbit. We prove that H₁f.o.(Σ, Q)^φ can be arbitrary large for any fixed φ ∈ Mod(Σ).