Let $\Lambda=R[G]$. Consider $L=Hom_R(M,N)$ as a $\Lambda$-module with the "conjugation" action: $(gf)(x)=gf(g^{-1}x)$. Let $I\subset \Lambda$ be the kernel of the augmentation $\Lambda\to R$, i.e. the ideal generated by $1-x$ for all $x\in G$. Then $A_n$ is the largest submodule of $L$ annihilated by $I^n$, or in other words $A=Hom_{\Lambda}(\Lambda/I^n,L)$.

I don't know what it's good for.

Let $\Lambda=R[G]$. Consider $L=Hom_R(M,N)$ as a $\Lambda$-module with the "conjugation" action: $(gf)(x)=gf(g^{-1}x)$. Let $I\subset \Lambda$ be the kernel of the augmentation $\Lambda\to R$, i.e. the ideal generated by $1-x$ for all $x\in G$. Then $A_n$ is the largest submodule of $L$ annihilated by $I^n$, or in other words $A_n=Hom_{\Lambda}(\Lambda/I^n,L)$.

I don't know what it's good for.