Linearly pushforwardable implies class-preserving for class-separating field

Statement
In a fact about::class-separating field, any fact about::linearly pushforwardable automorphism is class-preserving.

Class-separating field
A field $$k$$ is termed class-separating for a group $$G$$ if, given two elements $$g,h \in G$$ such that:

For every finite-dimensional linear representation $$\rho:G \to GL(V)$$, $$\rho(g)$$ and $$\rho(h)$$ are conjugate in $$GL(V)$$

Then, $$g$$ and $$h$$ are conjugate in $$G$$.

Class-preserving automorphism
An automorphism of a group is termed class-preserving if it sends every element of the group to an element in its conjugacy class.

Linearly pushforwardable automorphism
An automorphism $$\sigma$$ of a group $$G$$ is termed linearly pushforwardable over a field $$k$$ if, for any finite-dimensional linear representation $$\rho:G \to GL(V)$$, there exists an element $$a \in GL(V)$$ such that for every $$g \in G$$, we have:

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

Related facts

 * Class-preserving implies linearly pushforwardable in a class-determining field.

Related survey articles

 * Conjugacy class-representation duality

Proof
Given: A group $$G$$, a class-separating field $$k$$ for $$G$$. A linearly extensible automorphism $$\sigma$$ for $$G$$.

To prove: For any $$g \in G$$, $$g$$ and $$\sigma(g)$$ are conjugate.

Proof: Let $$\rho:G \to GL(V)$$ be any finite-dimensional linear representation of $$G$$ over $$k$$. Then, since $$\sigma$$ is linearly pushforwardable, the elements $$\rho(g)$$ and $$\rho(\sigma(g))$$ are conjugate inside $$GL(V)$$.

Since this is true for every finite-dimensional linear representation $$\rho$$, the definition of class-separating field forces us to conclude that $$g$$ and $$\sigma(g)$$ are conjugate.