We have a matrix representation $\phi$ of degree $d$ of a finite group $G$ over a field $F$, where character of $F$ doesn't divide $G$. From it we construct a representation $\psi : G \times G \to Aut(M_d)$ by setting $(g,h): X \to \phi(g)X\phi(h^{-1})$. I would like to show that $\psi$ is irreducible if $\phi$ is. I have some ideas but I got stuck and would like to see how to proceed.
To do this I wanted to use the space generated by coordinate functions of $\phi$, call it $V$. It is easy to show that this space is $\phi$-invariant. Now I assume that I have a $\psi$ invariant subspace $W$ and I would like to show that existence of $W$ implies existence of an $\phi$-invariant subspace of $V$. To do this I thought that maybe using the $\psi(g,1): X \to \phi(g)X$, because then for $w\in W$, this gives us that $\phi(g)w\in W$ but I don't think that this is very helpful. Either way I would appreciate any help with proving this.
The representation is isomorphic to the exterior tensor product $V\boxtimes V^\vee$, a $G\times G$-representation, by the standard isomorphism $$\begin{align*} V\otimes V^\vee&\xrightarrow\sim \hom(V,V)\\ v\otimes f&\mapsto (w\mapsto f(w)v). \end{align*}$$
This is proved to be irreducible here: Is the tensor product of irreducible representations of different groups irreducible?