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Restrict the quotient projection $\pi :G\to G/N$ to $H$. This is a group homomorphism withIts image is a nontrivial subgroup of the prime-order quotient group, because $H$ is not contained in $N$. By Lagrange's theorem for finite groups, the image has order $p$ and is all of $G/N$. The first isomorphism theorem for groups therefore gives

For the second assertion, partition $C$ into conjugacy classes for $N$. Because $N$ is normal, conjugation by $G$ permutes these smaller classes: $g{\mathrm{Cl}}_N(x)g^{-1}={\mathrm{Cl}}_N(gx{g}^{-1})$. This group action is transitive since any two elements of $C$ are conjugate in $G$. Elements of $N$ fix every $N$-class, so the action factors through $G/N$.

If $m$ is the number of smaller classes, the orbit-stabilizer theorem for this transitive action gives $m\mid |G/N|=p$. ThusDistinct conjugacy classes are disjoint because they are orbits of a group action. This proves conjugacy class splitting in a prime-index normal subgroup without assuming all $G$-conjugations are already conjugations by $N$.