Actually, for free abelian groups something does in fact happen along the lines of what you ask. It's just that the semidirect product idea is a little bit of a red herring.
In the $Z^n$ case, what happens is that any virtually $Z^n$ group surjects onto one of the finitely many $n$-dimensional Euclidean crystallographic groups, with finite kernel. This is basically the content of the Bieberbach theorems. In the rank $2$ case, that that's 17 different Wallpaper groups to surject onto. In rank $3$, that's 217 different Space groups.
