Quotient-pullbackable equals inner

This article gives a proof/explanation of the equivalence of multiple definitions for the term inner automorphism
View a complete list of pages giving proofs of equivalence of definitions
This fact is related to: Extensible automorphisms problem
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Statement

The following are equivalent for an automorphism $\sigma$ of a group $G$:

1. The automorphism is a quotient-pullbackable automorphism: For any homomorphism $\rho:H \to G$, there is an automorphism $\varphi$ of $H$, $\rho \circ \varphi = \sigma \circ \rho$.
2. The automorphism is an inner automorphism.

Definitions used

Quotient-pullbackable automorphism

An automorphism $\sigma$ of a group $G$ is termed quotient-pullbackable if given any surjective homomorphism $\rho: H \to G$ there is an automorphism $\varphi$ of $H$ such that $\rho \circ \varphi = \sigma \circ \rho$.

Inner automorphism

Further information: Inner automorphism

An automorphism $\sigma$ of a group $G$ is termed an inner automorphism if there exists $g \in G$ such that $\sigma = c_g = x \mapsto gxg^{-1}$.