Quotient-pullbackable automorphism

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quotient-pullbackable equals inner: A proof that every quotient-pullbackable automorphism of a group is an inner automorphism. An automorphism $σ$ of a group $G$ is termed a quoitent-pullbackable automorphism if for every group $K$ and surjective homomorphism $\rho:K \to G$ there exists an automorphism $σ'$ of $K$ such that $\rho \circ \sigma' = \sigma \circ \rho$ .
group property-conditionally quotient-pullbackable automorphism: The property of being quotient-pullbackable within the collection of groups satisfying a particular property.
variety-quotient-pullbackable automorphism
extensible automorphisms problem: A discussion of this and related problems, and the current best known results.