TITLE: If $G/N$ and $H/N$ are isomorphic, does it imply $G$ and $H$ are isomorphic?
QUESTION [8 upvotes]: If $N$ is a subgroup of both $G$ and $H$, and $G/N \cong H/N$, then is $G \cong H$?  (Or more generally, if $N$ and $M$ are subgroups of $G$ and $H$ respectively, and $N \cong M$ and $G/N \cong H/M$, then is $G \cong H$?)
I suspect the answer is no, but if the answer is yes, is there a natural isomorphism between $G$ and $H$ given an isomorphism $\phi:G/N \rightarrow H/N$?
Also, just making sure, is $(G \times H)/G \cong H$ and $(G \times H)/H \cong G$?

REPLY [4 votes]: Let $G=S_3$, $H=\mathbb Z_6$, $N_1=\langle(1~2~3)\rangle$ and $N_2=\langle[2]\rangle$. We have $$N_1\leq G,~~~N_2\leq\ H,~~~N_1\cong N_2$$ and $G/N_1\cong H/N_2$ but $G\ncong H$.