Normalizer of some subgroup of a group is necessarily normal subgroup of the group.

Question

Let $G$ be a group and $H$ be subgroup of $G$.

I already know that normalizer of H in G $N_G(H)$ is the largest subgroup of $G$ having H as a Normal subgroup.

So I showed that $N_G(N_G(H))=N_G(H)$. So I guess $N_G(H)$ is not necessarily normal of G.

Can you give me some example?