Normal core-closed subgroup property

Symbol-free definition
A subgroup property is said to be normal core-closed if whenever a subgroup has the property in the whole group, its normal core also has the property.

Definition with symbols
A subgroup property $$p$$ is said to be normal core-closed if whenever $$H$$ satisfies property $$p$$ in $$G$$, the normal core $$H_G$$ also satisfies $$p$$ in $$G$$.

Stronger metaproperties

 * Intersection-closed subgroup property
 * Finite-intersection-closed subgroup property when we are guaranteed that there are only finitely many conjugates
 * Conjugate-intersection-closed subgroup property

Related metaproperties

 * Normalizer-closed subgroup property
 * Normal closure-closed subgroup property
 * Characteristic core-closed subgroup property