Every saturated fusion system on a finite p-group is induced by a finite group containing it

Statement
Suppose $$P$$ is a group of prime power order with underlying prime $$p$$. Suppose $$\mathcal{F}$$ is a saturated fusion system on $$P$$. Then there exists a finite group $$G$$ containing $$P$$ such that the fusion system induced by $$G$$ on $$P$$ is precisely $$\mathcal{F}$$. (Does the result hold for fusion systems that aren't saturated? No idea).

Note that $$G$$ need not contain $$P$$ as a $$p$$-Sylow subgroup, even if $$\mathcal{F}$$ is a saturated fusion system. If $$\mathcal{F}$$ is a saturated fusion system and still cannot be induced from any finite group containing $$P$$ as a Sylow subgroup, then it is termed an exotic fusion system.