Assume that we have two groups $A$ and $B$ such that $C \subset A$ and $C \subset B$ where $C$ is a normal subgroup of both $A$ and $B$. If we have that $A/C \cong B/C$ is it true that $A \cong B$? I feel like this shouldn't be true but I can't seem to find any counter examples and my attempts to prove it thus far has been unsuccessful. It would be nice to find a finite counterexample if possible.
This is not the case where $A$ and $B$ have isomorphic subgroups $C$ and $C'$ such that $A/C \cong B/C'$. I have intentionally excluded these cases as uninteresting. Subgroups can be embedded in all sorts of strange ways into other groups. This forces the subgroup $C$ in $A$ to actually be the same set inside of $B$.