Hall-relatively weakly closed subgroup

Definition
Suppose $$G$$ is a finite group and $$H$$ is a subgroup of $$G$$. We say that $$H$$ is a Hall-relatively weakly closed subgroup of $$G$$ if, for any group $$L$$ containing $$G$$ as a defining ingredient::Hall subgroup, $$H$$ is a defining ingredient::weakly closed subgroup of $$G$$ relative to $$L$$.

Stronger properties

 * Weaker than::Isomorph-free subgroup
 * Weaker than::Sylow-relatively weakly closed subgroup: The same definition, but restricted to the case where $$G$$ is a group of prime power order.

Weaker properties

 * Stronger than::Coprime automorphism-invariant normal subgroup:
 * Stronger than::Coprime automorphism-invariant subgroup
 * Stronger than::Normal subgroup