Fusion system induced by a finite group on its p-Sylow subgroup is the inner fusion system iff the group is p-nilpotent

Statement

Suppose $G$ is a finite group and $P$ is a $p$-Sylow subgroup of $G$. Then, the fusion system induced by $G$ on $P$ (i.e., $\mathcal{F}_P(G)$) equals the inner fusion system $\mathcal{F}_P(P)$ if and only if $G$ is a P-nilpotent group (?), i.e., $G$ contains a Normal p-complement (?).

Facts used

1. Conjugacy-closed and Sylow implies retract

Proof

Given: A finite group $G$, a $p$-Sylow subgroup $P$ of $G$, such that $\mathcal{F}_P(G) = \mathcal{F}_P(P)$.

To prove: $G$ has a normal $p$-complement.

Proof:

1. Suppose $a,b \in P$ are conjugate in $G$. Then, $a,b$ are conjugate in $P$: Let $Q = \langle a \rangle, R = \langle b \rangle$. Then, if $gag^{-1} = b$ consider the map $\varphi:Q \to R$ given by $\varphi(x) = gxg^{-1}$. This is a morphism in $\mathcal{F}_P(G)$, so it is also a morphism in $\mathcal{F}_P(P)$. In particular, there exists $h \in P$ such that $hah^{-1} = b$.
2. $P$ has a normal complement in $G$, so $G$ has a normal $p$-complement: This follows from the previous step and fact (1).