Constant-weight codes (CWCs) have played an important role in coding theory. To construct CWCs, a K-GDD (where GDD is group divisible design) with the “star” property, denoted by K-*GDD, was introduced, in which any two intersecting blocks intersect in at most two common groups. In this paper, we consider the existence of 4-*GDDs. Previously, the necessary conditions for existence were shown to be sufficient for , and also sufficient for with prime powers and . We continue to investigate the existence of 4-*GDD(6n)s and show that the necessary condition for the existence of a 4-*GDD(6n), namely, , is also sufficient. The known results on the existence of optimal quaternary (n, 5, 4) CWCs are also extended.