Non-modular implies class-determining

Statement

Suppose $G$ is a finite group and $K$ is a field whose characteristic is either zero or a prime not dividing the order of $G$. Then, $K$ is a Class-determining field (?) for $G$: given two representations ${\phi }_1,{\phi }_2:G\to GL(n,K)$ such that ${\phi }_1(g)$ is conjugate to ${\phi }_2(g)$ for every $g\in G$, it is true that ${\phi }_1$ and ${\phi }_2$ are equivalent linear representations, i.e., there exists a matrix $A\in GL(n,K)$ satisfying $A{\phi }_1(g)A^{-1}={\phi }_2(g)$ for all $g\in G$.

Facts used

1. Non-modular implies character-determining: This basically says that the character determines the representation.
2. Character-determining implies class-determining: This says that if the character determines the representation, then knowing the conjugacy class of the image of every element suffices.

Proof

The proof follows from facts (1) and (2).