One of the main problems in design theory is to find construction methods to obtain unknown
designs. In this paper we give fundamental constructions of group divisible SBS designs with
block size 4 and 2 groups. We give necessary conditions for their existence and prove that these
conditions are sufficient. We use tools from graph theory and finite fields, for example-designs,
Steiner quadruple systems (SQSs), 1 and 2-factorisations of complete graphs.