Existence of a non-zero homomorphism
No. The map $\psi$ you provide is clearly $G$-invariant, but for distinct irreducible isomorphism types of $G$-modules, by Schur's lemma, there are no nonzero homomorphisms. The easiest example of this comes with $R=\mathbb R$, $G=\mathbb Z/(2)$, $M$ being $R$ with the trivial action, and $N$ being $R$ with the action by multiplication by $-1$.