Closure-characteristic subgroup

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

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

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