Linearly pushforwardable automorphism

Definition with symbols
Let $$G$$ be a group and $$k$$ be a field. An automorphism $$\sigma$$ of $$G$$ is termed linearly pushforwardable if for any linear representation $$\varphi:G \to GL_n(k)$$, there exists $$a \in GL_n(k)$$ such that for any $$g \in G$$, we have:

$$\varphi(\sigma(g)) = a\varphi(g)a^{-1}$$

Stronger properties

 * Weaker than::Inner automorphism (unconditionally)
 * Class-preserving automorphism when the field is a class-determining field (for instance, a field whose characteristic does not divide the order of the group, for a finite group)

Weaker properties

 * Linearly extensible automorphism
 * Class-preserving automorphism when the field is a class-separating field (for instance, a splitting field for a finite group)