Fully normalized potentially fully invariant subgroup

Definition
A subgroup $$H$$ of a group $$G$$ is termed a fully normalized potentially fully invariant subgroup of $$G$$ if it satisfies both these conditions:


 * 1) $$H$$ is a fully normalized subgroup of $$G$$, i.e., every automorphism of $$H$$ arises as the restriction to $$H$$ of an inner automorphism of $$G$$.
 * 2) $$H$$ is a potentially fully invariant subgroup of $$G$$, i.e., there exists a group $$K$$ containing $$G$$ such that $$H$$ is a fully invariant subgroup of $$K$$.