Grün's second theorem on the focal subgroup

Statement
Suppose $$G$$ is a finite group and $$p$$ is a prime number. Suppose further that $$G$$ is a fact about::p-normal group: in other words, the center of any $$p$$-Sylow subgroup is a fact about::weakly closed subgroup of Sylow subgroup, i.e., is weakly closed in it. Then, the fact about::focal subgroup of $$P$$ relative to $$G$$ equals the focal subgroup of $$P$$ relative to $$N_G(Z(P))$$.

By the focal subgroup theorem, this is the same as saying that:

$$P \cap [G,G] = P \cap [N_G(Z(P)), N_G(Z(P))]$$.

Related facts

 * Hall-Wielandt theorem