Suppose that $\pi:Y \to Y'$ is a regular covering of closed, connected, orientable three-manifolds and let $G$ be the deck transformation group. Furthermore, suppose that $Y$ is a rational homology sphere (I don't know how much this condition matters). It's not always the case that $G$ is solvable, since one can take $Y$ to be $S^3$ and $Y'$ to be the Poincare homology sphere. Are there other examples where $G$ is not solvable? Are such examples classified? If $Y$ has elliptic geometry, this is the only example (I think), but I have no clue in general. This seems like it could be related to residual finiteness of three-manifold groups/RFRS/LERF/other four-letter acronyms I don't understand.



When is a three-manifold deck transformation group solvable?
Cooper and Long showed [you can realize any finite group acting on a rational
homology sphere](http://dx.doi.org/10.1016/S0166-8641%2898%2900116-3). 


There's no general sort of classification that I know of. Maybe what you're asking for is,
given $Y'$ a rational homology sphere, and a homomorphism $\varphi: \pi\_1(Y')\to K$, where $K$ is a finite group,
when is the cover $Y\to Y'$ corresponding to $ker(\varphi)$ again a rational
homology sphere? One perspective is that this amounts to computing $H\_1(Y; \mathbb{Z}[K])$,
homology with twisted coefficients. [Dunfield and Thurston](http://www.ams.org/mathscinet-getitem?mr=1988291) took advantage of this
to compute covers of manifolds in the Snappea census with positive first betti number,
but such computations can get rapidly complicated, so I don't know of any general pattern when $K$ is non-solvable. 


