Existence of generalized Bhaskar Rao designs with block size 3

There are well-known necessary conditions for the existence of a generalized Bhaskar Rao design over a group GG, with block size k=3k=3. The recently proved Hall–Paige conjecture shows that these are sufficient when v=3v=3 and λ=|G|λ=|G|. We prove these conditions are sufficient in general when v=3v=3, and also when |G||G| is small, or when GG is dicyclic. We summarize known results supporting the conjecture that these necessary conditions are always sufficient when k=3k=3.