Symmetric 1-factorizations of the complete graph

Let S2nS2n be the symmetric group of degree 2n2n. We give a strong indication to prove the existence of a 1-factorization of the complete graph on (2n)!(2n)! vertices admitting S2nS2n as an automorphism group acting sharply transitively on the vertices. In particular we solve the problem when the symmetric group acts on 2p2p elements, for any prime pp. This provides the first class of GG-regular 1-factorizations of the complete graph where GG is a non-soluble group.