  First we study the Gorenstein cohomological dimension ${\rm Gcd}_RG$ of
groups $G$ over coefficient rings $R$, under changes of groups and rings; a
characterization for finiteness of ${\rm Gcd}_RG$ is given. Some results in
literature obtained over the coefficient ring $\mathbb{Z}$ or rings of finite
global dimension are generalized to more general cases. Moreover, we establish
a model structure on the weakly idempotent complete exact category
$\mathcal{F}ib$ consisting of fibrant $RG$-modules, and show that the homotopy
category $\mathrm{Ho}(\mathcal{F}ib)$ is triangle equivalent to both the stable
category $\underline{\mathcal{C}of}(RG)$ of Benson's cofibrant modules, and the
stable module category ${\rm StMod}(RG)$. The relation between cofibrant
modules and Gorenstein projective modules is discussed, and we show that under
some conditions such that ${\rm Gcd}_RG<\infty$, ${\rm Ho}(\mathcal{F}ib)$ is
equivalent to the stable category of Gorenstein projective $RG$-modules, the
singularity category, and the homotopy category of totally acyclic complexes of
projective $RG$-modules.
