F.g. groups with a finite index abelian subgroup It is well known that a virtually cyclic group is either finite, or finite-by-(infinite cyclic) or finite-by-(infinite dihedral).
I want to know if there is some similar description for f.g. virtually abelian groups, or even for simpler groups (e.g. virtually (free abelian of rank 2)).
In the affirmative case, are the proofs easy/short? References are welcome too.
Thank you!
 A: NOTE: SE does not allow me to comment this question, so I'm reporting (as an answer) my attempt to deal with a question which turns out to be equivalent to this one (you can find it here). Hope it's not a problem.  
A virtually $\mathbb{Z}^d$ group $G$ is also $\mathbb{Z}^d$-by-finite. Let us call $N$ a normal subgroup, of finite index in $G$, isomorphic to $\mathbb{Z}^d$.
We know that $Q = G/N$ is an arbitrary finite group and, in order to classify or characterize all such $G$, a possible approach could be made trying to solve the extension problem for $N$ and $Q$. What one have to do is to find, for all fixed $Q$, the homomorphisms $$\varphi\ :\ Q\ \rightarrow\ Aut(N)$$ in order to reconstruct the action of $Q$ on $N$. Since $Aut(\mathbb{Z}^d) \cong \mathsf{GL}(d,\mathbb{Z})$, one just has to consider the $k$-involutory elements of $\mathsf{GL}(d,\mathbb{Z})$.
It's easy to find some references with a classification of the finite subgroups of $\mathsf{GL}(d,\mathbb{Z})$ up to at least $d \le 7$ (and their number is finite $\forall\ d$) -if someone's interested, I can provide them.
Once we have such integer $k$-involutory matrices, we can hope to conclude something about $\varphi$ and then about $G$.
