A subgroup $H$ of $G$ is said to satisfy the Frattini Property if for any subgroup $K$ and $L$ such that $H\leq K \unlhd L$ implies that $L \leq N_L(H)K$

A subgroup is $H$ is pronormal in $G$ if for each $g \in G$, there exists $x \in \langle H, H^g \rangle$ such that $H^x = H^g$

A theorem characterising pronormal subgroups of soluble groups was proved by T. Peng 'Pronormality in Finite groups' which stated that: if $G$ is soluble group, $H$ is pronormal in $G$ $\iff$ H satisfies the Frattini Property. I do not have access to this paper nor does my institution have access to this

The $\Rightarrow$ direction is true in general since for any $g\in G$, $\langle H, H^g \rangle \leq H^{\langle g \rangle}$ and using my previous question https://math.stackexchange.com/questions/1810173/frattini-property-of-a-subgroup

For the $\Leftarrow$ direction, solvability of the group will be needed. I'm not sure how to proceed with proving this implication