Common circulant homogeneous factorisations of the complete digraph

In this paper we determine the positive integers nn and kk for which there exists a homogeneous factorisation of a complete digraph on nn vertices with kk ‘common circulant’ factors. This means a partition of the arc set of the complete digraph KnKn into kk circulant factor digraphs, such that a cyclic group of order nn acts regularly on the vertices of each factor digraph whilst preserving the edges, and in addition, an overgroup of this permutes the factor digraphs transitively amongst themselves. This determination generalises a previous result for self-complementary circulants.