Let $H$ be a Hall subgroup in a finite group G with $|H|=n$ (namely $gcd(n,|G:H|)=1$). Let $K$ be a subgroup of G with $|K|=n$, prove that $N_K(H)=H \cap K$. Moreover, show that $H$ is the unique subgroup of $N_G(H)$ with order $n$ .
Note: For the first question it's clear that $H \cap K \leq N_K(H)$, I attempted to show $|H \cap K| = |N_G(H) \cap K|$ by applying the formula $|AB|=|A||B|/|A \cap B|$ however I counld not proceed any further. For the second question I know if n is prime then by Sylow conjugate theorem one can show H is the unique sylow-n subgroup of $N_G(H)$, but I do not know how to prove for the case where n is composite.