Let $N$ be a normal subgroup of a finite group $G$ of prime index $p=|G: N|$.

By considering a suitable homomorphism, show that if $H$ is a subgroup of $G$ that is not contained in $N$, then $H \cap N$ is a normal subgroup of $H$ of index $p$.

Let $C$ be a conjugacy class of $G$ that is contained in $N$. Prove that $C$ is either a conjugacy class in $N$ or is the disjoint union of $p$ conjugacy classes in $N$.

[You may use standard theorems without proof.]