Primitive 2-factorizations of the complete graph

Let FF be a 2-factorization of the complete graph KvKv admitting an automorphism group G   acting primitively on the set of vertices. If FF consists of Hamiltonian cycles, then FF is the unique, up to isomorphisms, 2-factorization of KpnKpn admitting an automorphism group which acts 2-transitively on the vertex-set, see [A. Bonisoli, M. Buratti, G. Mazzuoccolo, Doubly transitive 2-factorizations, J. Combin. Designs 15 (2007) 120–132.]. 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.