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

Definition
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.

This satisfies the conditions necessary for being a saturated fusion system.

Note that we can define in a similar way the fusion system induced by a finite group on a finite p-subgroup. However, there is no guarantee in general that this category is a fusion system.

Facts

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