Quotient-pullbackable equals inner

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.

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

Related facts

 * Extensible implies inner
 * Finite-extensible implies inner
 * Finite-quotient-pullbackable implies inner