Suppose $N(H)$ is the normalizer of $H$ which is a subgroup of $G$.
If $H \lhd G$ or $H$ is a characteristic subgroup of $N(H)$, then $N(N(H))=N(H)$.
If $H \lhd G$, the result will be obvious. If $H$ is a characteristic subgroup of $N(H)$, then $H \lhd N(N(H))$ since $N(H) \lhd N(N(H))$. Hence, $N(N(H)) \subset N(H)$ and the result is followed.
Now, my question is under what other conditions $N(N(H))=N(H)$ does hold? That is when $H$ is not normal in $G$ and not characteristic in $N(H)$ but $N(N(H))=N(H)$.