Definition : A subgroup $H$ of $G$ is said to strongly pronormal in $G$, if for each $g\in G$ and for any subgroup $K \leq H$, there exists $x\in \langle H, K^g \rangle$ such that $K^{gx} \leq H$.
It is clear that every normal subgroup is strongly pronormal in $G$.
Let $G$ be a finite group. Then any maximal subgroup of $G$ is strongly pronormal in $G$.
Here is my attempt:
Let $M$ be a non-normal maximal subgroup of $G$. Then we must have that $M=N_G(M)$. Let $g \in G \setminus M$ and $K \leq M$. Since $M \leq \langle M, K^g \rangle \leq G$, we have that $M = \langle M, K^g \rangle$ or $G = \langle M, K^g \rangle$. In the first case, we get that $K^g =M$, and so $M$ is strongly pronormal in $G$.
For the case $G = \langle M, K^g \rangle$, I can't seem to draw a conclusion from this. Any help will be much appreciated.