pronormal subgroup

《Lecture Notes on Bayesian Estimation and Classification》 · 第 298 页 · Section 10.3「Frobenius's Criterion for p-Nilpotence」，Exercises 10.3，题 3

A subgroup $H$ is said to be pronormal in a group $G$ if for all $g$ in $G$ the subgroups $H$ and $H^g$ are conjugate in $\langle H,H^g\rangle$. If $H$ is pronormal and subnormal in $G$, show that $H\triangleleft G$.