Gorenstein analogues of a projectivity criterion over group algebras

We formulate and answer Gorenstein projective, flat, and injective analogues of a classical projectivity question for group rings under some mild additional assumptions. Although the original question, that was proposed by Jang-Hyun Jo in 2007, was for integral group rings, in this article, we deal with more general commutative base rings. We make use of the vast developments that have happened in the field of Gorenstein homological algebra over group rings in recent years, and we also improve and generalize several existing results from this area along the way.