Members of conjugation family that contain the p-core form a conjugation family

Statement
Suppose $$G$$ is a finite group, $$p$$ is a prime number, and $$\mathcal{F}$$ is a fact about::conjugation family for a $$p$$-Sylow subgroup $$P$$ of $$G$$. Then, the subset of $$\mathcal{F}$$ comprising those subgroups that contain $$O_p(G)$$ is also a conjugation family for $$P$$ in $$G$$.