Complete sets of metamorphoses: Paired stars into 4-cycles

If an edge-disjoint decomposition of a complete graph of order n into copies of a 3-star (i.e., the graph K on 4 vertices) is taken, and if these 3-stars can be paired up in three distinct ways to form a graph on 6 vertices consisting of a 4-cycle with two opposite pendant edges, such that: (1) in each of the three pairings, there exists a metamorphosis into a 4-cycle system; (2) taking precisely those 4-cycles formed from the two pendant edges from each pair of 3-stars, in each of the three metamorphoses, we again have a 4-cycle system of the complete graph, then this is called a complete set of metamorphoses from paired 3-stars into 4-cycles. We show that such a complete set of metamorphoses from paired 3-stars into 4-cycles exists if and only if the order of the complete graph is 1 or 9 (mod 24), and greater than 9