This article aims to contribute to the study of algebras with triangular decomposition over a Hopf algebra, as well as the BGG Category 𝒪. We study functorial properties of 𝒪 across various setups. The first setup is over a skew group ring, involving a finite group Γ acting on a regular triangular algebra A. We develop Clifford theory for A⋊Γ, and obtain results on block decomposition, complete reducibility, and enough projectives. 𝒪 is shown to be a highest weight category when A satisfies one of the “Conditions (S);” the BGG Reciprocity formula is slightly different because the duality functor need not preserve each simple module. Next, we turn to tensor products of such skew group rings; such a product is also a skew group ring. We are thus able to relate four different types of Categories 𝒪; more precisely, we list several conditions, each of which is equivalent in any one setup, to any other setup, and which yield information about 𝒪.