Core-characteristic subgroup

Symbol-free definition
A subgroup of a group is termed core-characteristic if its defining ingredient::normal core is a defining ingredient::characteristic subgroup of the whole group.

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is termed core-characteristic if the normal core $$H_G$$ of $$H$$ in $$G$$ is a characteristic subgroup of $$G$$.

Conjunction with other properties
Any normal subgroup that is also core-characteristic, is characteristic.

Incomparable properties

 * Closure-characteristic subgroup