Nonstandard definitions of inner automorphism

This page lists some offbeat, but correct, definitions of specific information about::inner automorphism.

Extensible automorphism
An automorphism $$\sigma$$ of a group $$G$$ is termed inner if it is an extensible automorphism: For any group $$K$$ containing $$G$$, $$\sigma$$ extends to an automorphism of $$K$$.

Pushforwardable automorphism
An automorphism $$\sigma$$ of a group $$G$$ is termed inner if it is a pushforwardable automorphism: For any homomorphism of groups $$\rho:G \to H$$, there is an automorphism $$\varphi$$ of $$H$$ such that $$\rho \circ \sigma = \varphi \circ \rho$$.

Quotient-pullbackable automorphism
An automorphism $$\sigma$$ of a group $$G$$ is termed inner if it is a quotient-pullbackable automorphism: For any surjective homomorphism $$\rho:K \to G$$, there exists an automorphism $$\sigma'$$ of $$K$$ such that $$\rho \circ \sigma' = \sigma \circ \rho$$.

I-automorphism
An automorphism of a group is termed an inner automorphism if it is an I-automorphism in the variety of groups.