Anosov flows, growth rates on covers and group extensions of subshifts

The aim of this paper is to study growth properties of group extensions of hyperbolic dynamical systems, where we do not assume that the extension satisfies the symmetry conditions seen, for example,
in the work of Stadlbauer on symmetric group extensions and of the authors on geodesic flows.
Our main application is to growth rates of periodic orbits for covers of an Anosov flow:
we reduce the problem of counting periodic orbits in an amenable cover $X$ to counting in a maximal abelian subcover $X^{\mathrm{ab}}$. In this way, we obtain an equivalence for the Gurevi\v{c} entropy: $h(X)=h(X^{\mathrm{ab}})$ if and only if the covering group is amenable.
In addition, when we project the periodic orbits for amenable covers $X$ to the compact factor $M$, they equidistribute with respect to a natural equilibrium measure -- in the case of the geodesic flow, the measure of maximal entropy.