Abelian-extensible automorphism-invariant subgroup

Definition
Suppose $$G$$ is an abelian group and $$H$$ is a subgroup of $$G$$. We say that $$H$$ is an abelian-extensible automorphism-invariant subgroup of $$G$$ if, for every defining ingredient::abelian-extensible automorphism $$\sigma$$ of $$G$$, we have $$\sigma(H) = H$$.