Suppose that $\pi:Y \to Y'$ is a regular covering of closed, connected, orientable threemanifolds 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 threemanifold groups/RFRS/LERF/other fourletter acronyms I don't understand.

Cooper and Long showed you can realize any finite group acting on a rational homology sphere. 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 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 nonsolvable. 


You are asking for free actions of finite, nonsolvable groups on $Y$. If you truly don't care that $Y$ is a rational homology sphere, for any finite group $G$ there exists a closed, connected, orientable 2manifold $X$ and a free action of $G$ on $X$, so $G$ also acts freely on $Y = X \times S^1$. 


Not exactly an answer to the question, but a classification of solvable groups which are fundamental groups of compact 3manifolds is given the following celebrated paper: Thomas, C. B. On 3manifolds with finite solvable fundamental group. Invent. Math. 52 (1979), no. 2, 187–197. It is very easy to check which of the manifolds produced are rational homology spheres... 

