Locally inner automorphism-balanced subgroup

Definition
A subgroup $$H$$ of a group $$G$$ is termed a locally inner automorphism-balanced subgroup if the following equivalent conditions are satisfied:


 * 1) Every inner automorphism of $$G$$ restricts to a locally inner automorphism of $$H$$.
 * 2) Every locally inner automorphism of $$G$$ restricts to a locally inner automorphism of $$H$$.