Pushforwardable automorphism

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• pushforwardable equals inner: Any pushforwardable automorphism is an inner automorphism. An automorphism $\sigma$ of a group $G$ is termed pushforwardable if, for any homomorphism $\rho :G\to H$, there exists an automorphism $\sigma '$ of $H$ such that $\rho \circ \sigma =\sigma '\circ \rho$.
• group property-conditionally pushforwardable automorphism
• variety-pushforwardable automorphism
• extensible automorphisms problem discusses this and related notions.