If $H\trianglelefteq K$ and $K\text{ char } G$, then is $H\trianglelefteq G$?

Question

Let $G$ be a finite group and $H,K$ be subgroups of $G$ such that $H$ is normal in $K$ and $K$ is characteristic in $G$. Is $H$ normal in $G$?

I know that if $H$ is characteristic in $K$ and $K$ is normal in $G$, then $H$ is normal in $G$. However, I was wondering if the above statement is true as well.

Answer 1

Not always. For a counterexample let $G=A_4$ and let $K$ be its Sylow $2$-subgroup. The Sylow $2$-subgroup is unique, so it is characteristic. In this case $K$ is abelian, so $H\le K$ implies $H\unlhd K$. But if $H$ is cyclic of order two, it is not a normal subgroup of $G$.