Fusion system induced by a finite group on its p-Sylow subgroup is a saturated fusion system

Statement
Suppose $$G$$ is a finite group, $$p$$ is a prime number, and $$P$$ is a $$p$$-Sylow subgroup of $$G$$. Suppose $$\mathcal{F}$$ is defined as a category with objects the subgroups of $$P$$ and morphisms given as follows: For every $$g \in G$$ and subgroups $$R,S \le P$$ such that $$gRg^{-1} \le S$$, there is a morphism $$\varphi:R \to S$$ given by $$\varphi(r) = grg^{-1}$$. This is a category on $$P$$ and is in fact a fusion system.

Proof that it is a category
First, we prove that we get a category on $$P$$.


 * It is closed under composition: Suppose $$\alpha:R \to S$$ and $$\beta:S \to T$$ are morphisms induced by conjugation by $$g,h \in G$$ respectively. Then, $$gRg^{-1} \le S$$ and $$hSh^{-1} \le T$$, so $$hgRg^{-1}h^{-1} \le T$$. Thus, there is a morphism in $$\mathcal{F}$$ from $$R$$ to $$T$$ given by conjugation by $$hg$$. We see that this is the composite $$\beta \circ \alpha$$.
 * All morphisms are group homomorphisms: This follows because each morphism arises as a restriction of an inner automorphism on the whole group.
 * It contains all inclusion maps: This follows because we can always set $$g$$ to be the identity element.
 * If $$\varphi:R \to S$$ is a morphism, then the restriction $$\varphi:R \to \varphi(R)$$, and its inverse, are morphisms. This follows from the definition.

Proof that it satisfies the conditions for a fusion system
We verify the three conditions:


 * All morphisms induced by inner automorphisms from $$P$$ are present: This is true, because in fact all morphisms induced by inner automorphisms from $$G$$ are present.
 * The inner automorphisms of $$P$$ form a $$p$$-Sylow subgroup of $$\operatorname{Aut}_{\mathcal{F}}(P)$$: $$\operatorname{Aut}_{\mathcal{F}}(P)$$ comprises the automorphisms induced by elements of $$N_G(P)$$. Since $$P$$ is $$p$$-Sylow in $$G$$, it is $$p$$-Sylow in $$N_G(P)$$, so since Sylow satisfies image condition, the image $$\operatorname{Aut}_P(P) = \operatorname{Inn}(P)$$ is $$p$$-Sylow in $$\operatorname{Aut}_{\mathcal{F}}(P)$$.
 * The extension axiom: