Isomorph-normal coprime automorphism-invariant of Sylow implies weakly closed

Statement
Suppose $$P$$ is a group of prime power order (where the underlying prime is $$p$$) and $$H$$ is an isomorph-normal coprime automorphism-invariant subgroup of $$P$$. In other words, we have the following:


 * $$H$$ is isomorph-normal in $$P$$: Any subgroup of $$P$$ isomorphic to $$H$$ is normal in $$P$$.
 * $$H$$ is coprime automorphism-invariant in $$P$$: Any $$p'$$-automorphism of $$P$$ leaves $$H$$ invariant.

Then, for any finite group $$G$$ containing $$P$$ as a $$p$$-Sylow subgroup, $$H$$ is weakly closed in $$P$$ with respect to $$G$$.

Related facts

 * Isomorph-normal characteristic of WNSCDIN implies weakly closed
 * Isomorph-containing iff weakly closed in any ambient group
 * Isomorph-normal coprime automorphism-invariant implies weakly closed for any fusion system

Facts used

 * 1) uses::Coprime automorphism-invariant normal subgroup of Hall subgroup is normalizer-relatively normal
 * 2) uses::Isomorph-normal implies conjugation-invariantly relatively normal in any ambient group
 * 3) uses::Sylow implies WNSCDIN
 * 4) uses::WNSCDIN implies every normalizer-relatively normal conjugation-invariantly relatively normal subgroup is weakly closed