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Projectively coresolved Gorenstein flat modules were introduced by Saroch andStovicek and were shown to be Gorenstein projective. We give characterizationsof Gorenstein projective, Gorenstein flat and projectively coresolvedGorenstein flat modules over a group ring $RG$, where $G$ is an$\textrm{LH}\mathfrak{F}$-group or a group of type $\Phi_R$ and $R$ is acommutative ring of finite Gorenstein weak global dimension. In this situation,we prove that every Gorenstein projective $RG$-module is projectivelycoresolved Gorenstein flat. We deduce that every Gorenstein projective$RG$-module is Gorenstein flat. The existence of weak characteristic modulesfor a group $G$ over a commutative ring $R$ plays a central role in ourresults. Furthermore, we determine the Gorenstein homological dimension of an$\textrm{LH}\mathfrak{F}$-group over a commutative ring of finite Gorensteinweak global dimension.

Gorenstein modules and dimension over large families of infinite groups