P-subgroup that is normal in every p-Sylow subgroup containing it

Definition
A $$p$$-subgroup $$K$$ of a group $$G$$ is termed a p-subgroup that is normal in every p-Sylow subgroup containing it if $$K$$ is a defining ingredient::normal subgroup inside every $$p$$-defining ingredient::Sylow subgroup of $$G$$ containing it.

Stronger properties

 * Weaker than::Normal p-subgroup
 * Weaker than::Sylow subgroup of normal subgroup
 * Weaker than::Normal subgroup of Sylow subgroup