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

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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.