2-subnormal quadratic offenders and Oliver's p-group conjecture

Bob Oliver conjectures that if $p$ is an odd prime and $S$ is a finite $p$-group, then the Oliver subgroup $\text{X}(S)$ contains the Thompson subgroup $J_e(S)$. A positive resolution of this conjecture would give the existence and uniqueness of centric linking systems for fusion systems at odd primes. Using ideas and work of Glauberman, we prove that if $p\ge 5$, $G$ is a finite $p$-group, and $V$ is an elementary abelian $p$-group which is an F-module for $G$, then there exists a quadratic offender which is 2-subnormal (normal in its normal closure) in $G$. We apply this to show that Oliver’s conjecture holds provided the quotient $G=S\mathrm{/}\text{X}(S)$ has class at most ${\log }_2(p-2)+1$, or $p \geq 5$and$G$ is equal to its own Baumann subgroup.