# Nonstandard definitions of inner automorphism

## In terms of being able to extend to bigger groups

### Extensible automorphism

Further information: Extensible equals inner, Inner implies extensible

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

Further information: Pushforwardable equals inner

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

Further information: Quotient-pullbackable equals inner, Inner implies quotient-pullbackable

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 $\rho \circ \sigma' = \sigma \circ \rho$.

## Definition in terms of universal algebra

### I-automorphism

Further information: Inner automorphisms are I-automorphisms in the variety of groups

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