Pushforwardable equals inner

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


 * 1) $$\sigma$$ is a pushforwardable automorphism of $$G$$: For any homomorphism $$\rho:G \to H$$, there exists an automorphism $$\varphi$$ of $$H$$ such that $$\rho \circ \sigma = \varphi \circ \rho$$.
 * 2) $$\sigma$$ is an inner automorphism.

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

Facts used

 * 1) uses::Extensible equals inner: An automorphism $$\sigma$$ of a group $$G$$ is inner if and only if it can be extended to an automorphism for any group containing $$G$$.

Proof that inner implies pushforwardable
If $$\sigma = c_g$$ for $$g \in G$$, and $$\rho:G \to H$$ is a homomorphism, we can take $$\varphi = c_{\sigma(g)}$$.

Proof that pushforwardable implies inner
If $$\sigma$$ is pushforwardable, then, in particular, $$\sigma$$ is extensible in the sense that it can be extended to an automorphism for any group containing $$G$$. Thus, by fact (1), $$\sigma$$ is inner.