If $H$ is a subgroup of $K$ and $H$ and $K$ are normal subgroups of $G$, must $H$ be a normal subgroup of $K$?

Question

Suppose $H$ and $K$ are normal subgroups of a group $G$, and also $H$ is a subgroup of $K$. Must $H$ be a normal subgroup of $K$?

Answer 1

$gHg^{-1}=H$ for all $g\in G$. In particular this is true for all $g\in K$.

The assumption that $K$ is normal is not needed. If $H$ is normal in $G$, it is also normal in any subgroup of $G$ containing it.