Let G be a group and H be a subgroup. Suppose N is a normal subgroup of G that is contained in H, and that $G/N\cong H/N$. Does this imply that $G\cong H$?
If G is finite then $G/N\cong H/N$ obviously implies that $G=H$, so only the inifinite case is to be considered. I have tried to find a counterexample (since this proposition doesn't look true), but haven't been able to.