  Let S be an orientable surface with negative Euler characteristic, let
\psi\in\Mod(S) be a mapping class of S, and let T_{\psi} be the mapping torus
of \psi. We study the action of lifts of \psi on the homology of finite covers
of S via the torsion homology growth of towers of finite covers of T_{\psi}. We
show that \psi admits a lift to a finite cover with a homological eigenvalue of
length greater than one if and only if the mapping torus T_{\psi} admits a
finite cover X and a certain tower of abelian covers which have exponential
torsion homology growth. We show that the existence of such a lift of \psi is
intrinsic to T_{\psi}, in the sense that it does not depend on the particular
fibration used to present T_{\psi}.
