$F$-stability in finite groups

Let $G$ be a finite group, $S \in \mathit {Syl}_p(G)$, and $\mathcal S$ be the set subgroups containing $S$. For $M \in \mathcal S$ and $V = \Omega_1Z(O_p(M))$, the paper discusses the action of $M$ on $V$. Apart from other results, it is shown that for groups of parabolic characteristic $p$ either $S$ is contained in a unique maximal $p$-local subgroup, or there exists a maximal $p$-local subgroup in $M \in \mathcal S$ such that $V$ is a nearly quadratic 2F-module for $M$.