Let me start this question with an example that hopefully makes clear what I am looking for:

A discrete subgroup $G$ of the group of euclidean isometries of $\mathbb{R}^d$ is called a crystallographic group if it contains a lattice $L$ of full rank. In this case, $G/L \cong P$ is a finite crystallographic subgroup of the orthogonal group. So what we get is a short exact sequence $$0 \rightarrow \mathbb{Z}^d \rightarrow G \rightarrow P \rightarrow 1,$$ which can also be expressed as stating that $G$ is isomorphic to an extension of $\mathbb{Z}^d$ by $P$. Remarkably, the converse is also true. Namely that any such extension is isomorphic to a crystallographic group. So the family of crystallographic groups is in $1$-to-$1$ correspondence with group extensions of $\mathbb{Z}^d$ by a crystallographic point group.

Now, for my question. What other interesting examples of families of groups are there that can also be described by group extensions in a similar way as crystallographic groups can?

Let me add a few sentences to maybe prevent misinterpretation of my question. I am certainly aware of the theorem of Jordan-Hölder and its consequence that one in principle could survey all finite groups by using the classification of all finite simple groups and solving the extension problem. But this also means allowing for very complicated sets of groups that can be used as a first part and last part in the above sequence. I am looking for more reasonable examples, where "reasonable" appeals to everyone's intuition.