Let $G$ be a finite group. Let $H \leq K\unlhd G$. If for each $P$ Sylow subgroup of G there exists $x \in G$ such that $HP^{x}=P^{x}H$ then for each $P \cap K$ of $K$ there exists $y \in K$ such that $H(P\cap K)^{y}=(P\cap K)^{y}H$.
I know that $H(P\cap K)^{x}=(P\cap K)^{x}H$, but $x$ may not be an element of $K$. How may I prove this?