  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.
