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

This article gives the statement and possibly, proof, of an implication relation between two subgroup properties, when the big group is a group of prime power order. That is, it states that in a Group of prime power order (?), every subgroup satisfying the first subgroup property (i.e., Isomorph-normal coprime automorphism-invariant subgroup (?)) must also satisfy the second subgroup property (i.e., Sylow-relatively weakly closed subgroup (?)). In other words, every isomorph-normal coprime automorphism-invariant subgroup of group of prime power order is a Sylow-relatively weakly closed subgroup of group of prime power order.
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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. Coprime automorphism-invariant normal subgroup of Hall subgroup is normalizer-relatively normal
2. Isomorph-normal implies conjugation-invariantly relatively normal in any ambient group
3. Sylow implies WNSCDIN
4. WNSCDIN implies every normalizer-relatively normal conjugation-invariantly relatively normal subgroup is weakly closed

Proof

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