A functorial approach to modules of G-dimension zero

Let $R$ be a commutative Noetherian ring and let $\G$ be the category of modules of G-dimension zero over $R$. We denote the associated stable category by $\pG$. We show that the functor category $\modpG$ is a Frobenius category and we argue how this property could characterize $\G$ as a subcategory of $\modR$.