# Fusion system induced by a finite group on its p-Sylow subgroup

Suppose $G$ is a finite group, $p$ is a prime number, and $P$ is a $p$-Sylow subgroup. The fusion system on $P$ induced by $G$ is defined as follows: for every element $g \in G$, 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 fusion system is often written as $\mathcal{F}_P(G)$. The special case where $G = P$ gives rise to what we call the inner fusion system.