Abelian-quotient-pullbackable automorphism

Definition
Suppose $$G$$ is an abelian group and $$\sigma$$ is an automorphism of $$G$$. We say that $$\sigma$$ is an abelian-quotient-pullbackable automorphism of $$G$$ if, for any surjective homomorphism $$\rho:K \to G$$ from an abelian group $$K$$, there exists an automorphism $$\sigma'$$ of $$K$$ such that $$\rho \circ \sigma' = \sigma \circ \rho$$.

Related properties

 * Abelian-quotient-pullbackable endomorphism
 * Abelian-extensible automorphism
 * Abelian-extensible endomorphism

Facts

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