Abelian-extensible automorphism

Symbol-free definition
An automorphism of an abelian group is termed abelian-extensible if it can be extended to an automorphism for any embedding of the group in an abelian group.

Definition with symbols
An automorphism $$\sigma$$ of an Abelian group $$G$$ is termed abelian-extensible if, for any embedding of $$G$$ as a subgroup of an abelian group $$H$$, there exists an automorphism $$\sigma'$$ of $$H$$ whose restriction to $$G$$ equals $$\sigma$$.

Other related properties

 * Abelian-extensible endomorphism

Facts

 * Divisible abelian group implies every automorphism is abelian-extensible
 * Abelian-extensible automorphism not implies power map
 * Subgroup of abelian group not implies abelian-extensible automorphism-invariant

Related facts

 * Divisible abelian group implies every endomorphism is abelian-extensible
 * Free abelian group implies every automorphism is abelian-quotient-pullbackable
 * Free abelian group implies every endomorphism is abelian-quotient-pullbackable