Primitive 2-factorizations of the complete graph

Let F be a 2-factorization of the complete graph Kv admitting an automorphism group G acting primitively on the set of vertices. If F consists of Hamiltonian cycles, then F is the unique, up to isomorphisms, 2-factorization of Kpn admitting an automorphism group which acts 2-transitively on the vertex-set. In the non-Hamiltonian case we construct an infinite family of examples whose automorphism group does not contain a subgroup acting 2-transitively on vertices