If $G, G'$ are two groups whose categories of finite-indexed normal subgroups are equivalent. Then, are they or their profinite completions isomorphic ?
If instead G, G' are topological groups (perhaps compact or locally compact) with equivalent categories of finite-index closed subgroups, then can we also conclude that $G \cong G'$ ?
This fails quite severely.
For any prime $p$ and any positive integer $n$, the poset of finite-index normal subgroups of $\mathbb{Z}/p^n\mathbb{Z}$ is isomorphic to the ordinal $n+1 = \{0, \dots, n\}$. And these groups are all finite, so they are canonically isomorphic to their profinite completions. You can also view them as discrete topological groups, in which case they are compact and Hausdorff.
By varying $p$, we get an infinite collection of pairwise non-isomorphic finite (compact Hausdorff topological) groups with the same poset of finite-index normal subgroups.
By varying $n$, we get an infinite collection of posets which admit such an infinite collection of finite groups.