Control of fusion is local

Statement
Suppose $$G$$ is a finite group, and $$p$$ is a prime number. Suppose $$W$$ is a conjugacy functor on the $$p$$-subgroups of $$G$$. Then, if $$W$$ controls fusion in every p-local subgroup (i.e., the normalizer of every non-identity $$p$$-subgroup), then $$W$$ controls fusion in $$G$$.

Facts used

 * 1) uses::Alperin's fusion theorem in terms of well-placed tame intersections
 * 2) uses::Subsets of a well-placed tame intersection that are conjugate in its normalizer are conjugate in the normalizer of the conjugacy functor

Proof
(This proof uses the right-action convention)

Given: A finite group $$G$$, a prime number $$p$$. A conjugacy functor $$W$$ on the $$p$$-subgroups of $$G$$, such that $$W$$ controls fusion in $$N_G(H)$$ for every nontrivial $$p$$-subgroup $$H$$ of $$G$$.

To prove: $$W$$ controls $$p$$-fusion in $$G$$. In other words, given a $$p$$-Sylow subgroup $$P$$ and two subsets $$A,B \subseteq P$$ that are conjugate in $$G$$, there exists $$g \in N_G(W(P))$$ such that $$A^g = B$$.

Proof: By fact (1), there exist well-placed tame Sylow intersections $$P \cap Q_i$$ for $$1 \le i \le n$$, and elements $$y_i \in N_G(P \cap Q_i)$$ such that $$A_0 = A$$, $$A_i^{y_i} = A_{i+1}$$, and $$A_n = B$$. By fact (2), we obtain that in fact, there exist $$z_i \in N_G(W(P))$$ such that $$A_i^{z_i} = A_{i+1}$$. Thus, the element $$z = z_1z_2 \dots z_n$$ is an element of $$N_G(W(P))$$ that conjugates $$A$$ to $$B$$.