My professor recently mentioned (when asked how to do a homework problem in office hours) the following technique for finding subgroups of $\mathbb Z \times \mathbb Z$: consider the short exact sequence $$ 1 \to \mathbb Z \xrightarrow{i_1} \mathbb Z \times \mathbb Z \xrightarrow{\pi_2} \mathbb Z \to 1 $$ where $i_1$ is inclusion into the first factor and $\pi_2$ is projection onto the second factor. Every subgroup of $\mathbb Z \times \mathbb Z$ can be found by looking at subgroups of the copies of $\mathbb Z$ and seeing what $\mathbb Z \times \mathbb Z$ can restrict to and get another short exact sequence; that is, you look at $$ 1 \to m \mathbb Z \to A \to n \mathbb Z \to 1 $$ where the maps are restrictions of the maps from above and see which $A$ are possible, and if you look at every possible subgroup of $\mathbb Z$, which is to say every possible $m$ and $n$, then you'll find all the possible subgroups.
He then extended this technique to $\mathbb Z \ltimes \mathbb Z$, where $\mathbb Z$ is acting on $\mathbb Z$ by $ k \cdot n = (-1)^k n$.
I sort of get how to do it in practice, but I definitely don't really understand why this works. Does it work for any group? Just products? Is there anyone who knows what I'm talking about (since I may have explained it incorrectly) that could clarify and explain?
(For context, this was in an algebraic topology course; the original question was to classify the covering spaces of the torus and of the Klein bottle, but this is the same as classifying subgroups of their fundamental groups, namely $\mathbb Z \times \mathbb Z$ and $\mathbb Z \ltimes \mathbb Z$.)