Abelian-potentially characteristic subgroup

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Definition

A subgroup $H$ of an abelian group $G$ is termed an abelian-potentially characteristic subgroup if there exists an abelian group $K$ containing $G$ such that $H$ is a characteristic subgroup of $K$.

Relation with other properties

Stronger properties

property	quick description	proof of implication	proof of strictness (reverse implication failure)	intermediate notions
Characteristic subgroup of abelian group	characteristic subgroup of abelian group	direct from definition	abelian-potentially characteristic not implies characteristic	|FULL LIST, MORE INFO
Abelian-potentially fully invariant subgroup	is a fully invariant subgroup in some bigger abelian group	follows from fully invariant implies characteristic	?	|FULL LIST, MORE INFO
Abelian-potentially verbal subgroup	is a verbal subgroup in some bigger abelian group			|FULL LIST, MORE INFO
Subgroup of finite abelian group		finite abelian NPC theorem		Abelian-potentially verbal subgroup|FULL LIST, MORE INFO

Weaker properties

property	quick description	proof of implication	proof of strictness (reverse implication failure)	intermediate notions
Abelian-extensible automorphism-invariant subgroup	invariant under all abelian-extensible automorphisms	abelian-potentially characteristic implies abelian-extensible automorphism-invariant	?	|FULL LIST, MORE INFO
Subgroup of abelian group	subgroup of an abelian group			Abelian-extensible automorphism-invariant subgroup|FULL LIST, MORE INFO