Is group extension unique?

Question

Definition: Given groups N and H, a group is said to be an extension of H by N if there exists $N_{0}\vartriangleleft G$ such that $N_{0}\cong N$ and $G/N_{0}\cong H$ We know that a normal subgroup of a group is not unique. Then extension of a group is not uniquely determined. But I dont find example. Can you give me an example for that $N_0$ is not unique?

Answer 1

$\mathbb Z_4\times \mathbb Z_2$ is an extension of $\mathbb Z_4$ by $\mathbb Z_2$ as well as an extension of $\mathbb{Z}_2\times \mathbb Z_2$ by $\mathbb Z_2$. For the first, $N_0$ is generated by $(0,1)$, and for the second it is generated by $(2,0)$.

If you want to keep $H$ the same, let's go back to the $\mathbb{Z}_4$ by $\mathbb{Z}_2$ example. You get the same isomorphism class of extensions if you choose the subgroup generated by $(2,1)$.