It is the exercise 2.12 on Paolo Aluffi's Algebra: Chapter 0.
Let $P$ be a p-Sylow subgroup of $G$ and $H$ a subgroup containing $N_G(P)$. It claims that $p|([G:H]-1)$.
I tried to use the fact that $H$ is self-normalizing, but could not see any relevance to this exercise. I also tried to imitate the trick which proved Sylow-III but failed.