Class-preserving implies linearly pushforwardable

Statement
Suppose $$G$$ is a group and $$k$$ is a fact about::class-determining field for $$G$$. Then, any fact about::class-preserving automorphism of $$G$$ is linearly pushforwardable.

Class-determining field
A field $$k$$ is termed class-determining for a group $$G$$ if, given any two finite-dimensional linear representations of $$G$$ on a vector space $$V$$ over $$k$$, say $$\rho_1, \rho_2: G \to GL(V)$$, such that for every $$g \in G$$, the elements $$\rho_1(g)$$ and $$\rho_2(g)$$ are conjugate inside $$GL(V)$$, we can conclude that $$\rho_1$$ and $$\rho_2$$ are equivalent.

In other words, the conjugacy classes in $$GL(V)$$ of the images of elements in $$G$$, determine the representation.

Note that what this statement really says is that if two representations are conjugate at every element of $$G$$, they are equivalent, or globally conjugate.

For a finite group, any field whose characteristic does not divide the order of the group is a character-determining field, and hence a class-determining field.

Class-preserving automorphism
An automorphism of a group is termed class-preserving if it sends each element to within 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 if $$c_a$$ denotes conjugation by $$a$$, then $$\rho \circ \sigma = c_a \circ \rho$$. In other words, for any $$g \in G$$:

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

Related facts

 * Linearly pushforwardable implies class-preserving in a class-separating field

Related survey articles

 * Conjugacy class-representation duality

Proof
Given: A group $$G$$, a class-determining field $$k$$ for $$G$$, a class-preserving automorphism $$\sigma$$ of $$G$$, and a finite-dimensional linear representation $$\rho: G \to GL(V)$$

To prove: There exists $$a \in GL(V)$$ such that $$\rho(\sigma(g)) = a \sigma(g)a^{-1}$$ for every $$g$$.

Proof: Observe first that $$\sigma$$ and $$\rho \circ \sigma$$ are both linear representations of $$G$$, since $$\sigma$$ is an automorphism. Further, since $$\sigma$$ is class-preserving, it is true that for any $$g \in G$$, there exists $$h \in G$$ such that $$\sigma(g) = hgh^{-1}$$. We thus obtain:

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

In other words, $$\rho(\sigma(g))$$ and $$\rho(g)$$ are conjugate in $$GL(V)$$.

Now, by the definition of class-determining field, we see that $$\rho \circ \sigma$$ and $$\rho$$ are equivalent linear representations. Thus, there exists $$a \in GL(V)$$ such that for every $$g \in G$$:

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