Characteristic core-closed subgroup property

Symbol-free definition
A subgroup property is said to be characteristic core-closed if whenever a subgroup has the proeprty in the whole group, so does its characteristic core (viz the largest characteristic subgroup contained inside it).

Definition with symbols
A subgroup property $$p$$ is said to be characteristic core-closed if whenever $$H$$ satisfies property $$p$$ in $$G$$, and $$K$$ is the characteristic core of $$H$$, $$K$$ satisfies property $$p$$ in $$G$$.

Stronger metaproperties

 * Intersection-closed subgroup property

Related metaproperties

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