Sylow-relatively strongly closed subgroup

Definition
Suppose $$P$$ is a group of prime power order and $$H$$ is a subgroup of $$P$$. We say that $$H$$ is a Sylow-relatively strongly closed subgroup if, for any finite group $$G$$ containing $$P$$ as a Sylow subgroup, $$H$$ is a defining ingredient::strongly closed subgroup of $$P$$ with respect to $$G$$.

Stronger properties

 * Weaker than::Subisomorph-containing subgroup of group of prime power order

Weaker properties

 * Stronger than::Sylow-relatively weakly closed subgroup
 * Stronger than::Normal subgroup of group of prime power order