The following is a question asked to me these days by Gülin Ercan.

Let $G = L(q^f)$ be a finite simple group of Lie type, and let $L(q) \cong H \le G$ be the group of fixed points of the automorphisms of $G$ induced by field automorphisms. In case $H$ is nonsolvable, can there exist a nontrivial subgroup $N$ of $G$ which is normalized by $H$ and which has trivial intersection with $H$?