Symmetric 1-factorizations of the complete graph

Let S_2n be the symmetric group of degree 2n. We give a strong indication to prove the existence of a 1-factorization of the complete graph on (2n)! vertices admitting S_2n as an automorphism group acting sharply transitively on the vertices. In particular we solve the problem when the symmetric group acts on 2p elements, for any prime p. This provides the first class of G-regular 1-factorizations of the complete graph where G is a non-soluble group