$ZJ$-theorems for fusion systems

For $p$ an odd prime, we generalise the Glauberman-Thompson $p$-nilpotency theorem (Gorenstein, 1980) to arbitrary fusion systems. We define a notion of $Qd(p)$-free fusion systems and show that if $\mathcal{F}$ is a $Qd(p)$-free fusion system on some finite $p$-group $P$, then $\mathcal{F}$ is controlled by $W(P)$ for any Glauberman functor $W$, generalising Glauberman's $ZJ$-theorem (Glauberman, 1968) to arbitrary fusion systems.