Is a Subgroup Characteristic in its Normalizer?

Question

Let $G$ be a finite group and $H \subseteq G$. Is it true that $H$ is a characteristic subgroup of $N_{G}(H)$? Knowing that "the something" subgroup must be characteristic, I believe it must be true. Any comments would be appreciated!

Answer 1

Let $G=C_p\times C_p$ and $H$ any subgroup of order $p$. Then $N_G(H)=G$ but $H$ is not characteristic in $G$.

Answer 2

The normalizer of any subgroup of an abelian group is the whole group, but there are subgroups of abelian groups that are not characteristic.

Answer 3

There is also a positive answer to the question.

If $H \leq G$ with gcd$(|H|, |G:H|)=1$, then $H$ char $N_G(H)$. (Example are Sylow subgroups!). The statement follows from the fact a normal subgroup having coprime order and index, must be characteristic.