This is a general question inspired by a post I read on MSE. Let $A, B, C, D$ be groups, $C, D$ normal subgroups of $A, B$ respectively. If $A \cong B$, $\frac{A}{C} \cong \frac{B}{D}$, what conditions do we need so that $C \cong D$? They can be finite or infinite groups.
We know this isn't true in general, as seen in the second part of this question.
It felt natural to try to use some of the four group isomorphism theorems, but I couldn't seem to make them work in this case.