isomorphism of quotient groups

Question

If G1 and G2 are groups ,H1 and H2 are normal subgroups,and G1/H1 and G2/H2 are quotient groups, then Is this true: If G1/H1 and G2/H2 ,and H1 and H2 are isomorph then G1 and G2 are isomorph too. I need an example to show it's not true. Thanks.

Answer 1

You could take $G_1=S_3$, $H_1=A_3$ where $S_3$ denotes permutation group on $\{1,2,3\}$ and $A_3$ the even permuations in $S_3$.

Next to that take $G_2=\mathbb Z_6$, and $H_2=\{\bar0,\bar2,\bar4\}$.

The subgroups and quotients all have prime order, hence are cyclic.

$\mathbb Z_6$ is abelian, but $S_3$ is not, so they are not isomorphic.