$\aleph_m$-presented Gorenstein flat modules and Gorenstein dimensions of groups

Let $G$ be a group and $k$ be a commutative ring. The Gorenstein homological dimension $\text{Ghd}_k G$ and the Gorenstein cohomological dimension $\text{Gcd}_k G$ of $G$ over $k$ are defined as the Gorenstein flat and the Gorenstein projective dimension, respectively, of the trivial $kG$-module $k$