Alperin's fusion theorem in terms of tame intersections

Statement
Suppose $$G$$ is a finite group, $$p$$ is a prime, and $$P$$ is a $$p$$-Sylow subgroup of $$G$$. Then, the collection of tame Sylow intersections involving $$P$$ form a conjugation family for $$P$$ in $$G$$.

Explicit statement using the right-action convention
$$G$$ is a finite group, $$p$$ is a prime, and $$P$$ is a $$p$$-Sylow subgroup of $$G$$. Suppose $$A,B$$ are subsets of $$P$$ that are conjugate by some element $$g \in G$$. Then, there exists a collection of fact about::tame Sylow intersections $$P \cap Q_i, 1 \le i \le n$$ and a collection of elements $$g_i \in N_G(P \cap Q_i)$$ such that:


 * $$\langle A \rangle \le P \cap Q_1$$.
 * $$A^{g_1g_2 \dots g_r} \le P \cap Q_{r+1}$$ for any $$1 \le r \le n - 1$$.
 * $$g = g_1g_2 \dots g_n$$.

Related facts

 * Alperin's fusion theorem in terms of conjugation families: This is a somewhat weaker version.
 * Alperin's fusion theorem in terms of well-placed tame intersections: This is a somewhat stronger version.