Let $G$ be a group and $H$ a finite index subgroup, one could study the cardinality of the set of subgroups $K$ of $H$ of fixed finite index $[H:K]=n$.
At first I thought this was always a finite set, but I soon hit on an example where it is not so, e.g. the direct product of infinite copies of $\mathbb{Z}$, where we can take $G=H$ and we have infinite different subgroups $K$ of index 2, by choosing $K$ as the product of $2 \mathbb Z \times \prod \mathbb Z$ where $2 \mathbb Z$ is each time the subgroup of a different copy of $\mathbb Z$.
So my question is: How do we characterize groups $G$ that has the following property
"For every finite index subgroup $H$, and every fixed $n>0$ there is only a finite number of subgroups $K$ of $H$ such that $[H:K]=n$?
Note for example that every finitely generated abelian group $G$ has this property, thanks to the classification theorem: every subgroup $H$ of $G$ is of the form $H_{\text {free}} \bigoplus H_{\text{ torsion}}$, so asking it is of finite index means that the ranks of $G$, $H$ and $K$ are the same. But then $H_{\text{torsion}}$ is finite, thus it has finitely many subgroups.
I don't know if this is a trivial question or an uninteresting one, so any reference or comment is welcome. Thanks!