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 $G$.