On 2-factorizations whose automorphism group acts doubly transitively on the factors

Let FF be a 2-factorization of the complete graph KvKv admitting an automorphism group G   acting doubly transitively on the set of factors. In the Hamiltonian case the only possibility is the unique factorization of K5K5, while in the non-Hamiltonian case we give some classes of examples and some necessary conditions for the existence of such factorizations.