Normal Sylow $p$-subgroup of a normal subgroup

Question

Any hints for the following question - I am sure that I am missing something very simple here.

$K$ is a normal subgroup of $G$ and $P$ is a Sylow $p$-subgroup of $K$. If $P$ is a normal subgroup of $K$, show that it is also a normal subgroup of $G$.

$G$ is a finite group

Answer 1

Hint: If $g \in G$, then $g^{-1}Pg \leq K$ since $K$ is a normal subgroup.