Abelian-quotient-pullbackable automorphism-invariant subgroup

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Definition

A subgroup $H$ of an abelian group $G$ is termed an abelian-quotient-pullbackable automorphism-invariant subgroup if, for every abelian-quotient-pullbackable automorphism $\sigma$ of $G$, $\sigma$ restricts to an automorphism of $H$.

Formalisms

Function restriction expression

This subgroup property is a function restriction-expressible subgroup property: it can be expressed by means of the function restriction formalism, viz there is a function restriction expression for it.
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Function restriction expression	$H$ is a fully invariant subgroup of $G$ if ...	This means that full invariance is ...	Additional comments
abelian-quotient-pullbackable automorphism $\to$ function	every abelian-quotient-pullbackable automorphism of $G$ sends every element of $H$ to within $H$	the invariance property for abelian-quotient-pullbackable automorphisms
abelian-quotient-pullbackable automorphism $\to$ endomorphism	every abelian-quotient-pullbackable automorphism of $G$ restricts to an endomorphism of $H$	the endo-invariance property for abelian-quotient-pullbackable automorphisms; i.e., it is the invariance property for abelian-quotient-pullbackable automorphism, which is a property stronger than the property of being an endomorphism
abelian-quotient-pullbackable automorphism $\to$ automorphism	every abelian-quotient-pullbackable automorphism of $G$ restricts to an automorphism of $H$	the auto-invariance property for abelian-quotient-pullbackable automorphisms; i.e., it is the invariance property for abelian-quotient-pullbackable automorphism, which is a group-closed property of automorphisms