Elementary abelian of prime-square order implies corresponding prime field is not class-determining

Statement
Suppose $$p$$ is a prime number and $$G$$ is an elementary abelian group of prime-square order, i.e., it is the external direct product of two copies of the group of prime order. Let $$K$$ be the prime field of characteristic $$p$$, i.e., the field $$\mathbb{F}_p$$.

Then, $$K$$ is not a class-determining field for $$G$$. Specifically, it is possible to construct two three-dimensional representations $$\varphi_1, \varphi_2: G \to GL(3,K)$$ with the property that $$\varphi_1(g)$$ and $$\varphi_2(g)$$ are conjugate for every $$g \in G$$, but $$\varphi_1$$ and $$\varphi_2$$ are not equivalent linear representations.

Related facts

 * Cyclic implies every field is class-determining
 * Character determines representation in characteristic zero
 * Non-modular implies class-determining