Isomorphic normal subgroups

Question

Let $N_1$ and $N_2$ be normal subgroups of G so that $N_1$ and $N_2$ are isomorphic. Is it true that then also $G/N_1$ is isomorphic to $G/N_2$?

Answer 1

No. Let $N_1= 3 \mathbb{Z}$, $N_2 =4 \mathbb{Z}$. Both are normal subgroups of the additive group $\mathbb{Z}$. They are isomorphic via $f: 3x \mapsto 4x$. However, $\mathbb{Z}/3 \mathbb{Z}$ is not isomorphic to $\mathbb{Z}/ 4 \mathbb{Z}$ since these are both finite groups with a different number of elements, hence no bijection exists between them.

Answer 2

No. Consider $G=Z_2\times Z_4$. This is abelians so all subgroups are normal. Try quotienting by some of the copies of $Z_2$ you can find in this group.