Vertex transitive embeddings of complete graphs

It is shown that a vertex transitive complete map M satisfies one of the following: (i) Aut M is regular on the vertex set, (ii) Aut M has a subgroup of index at most 2 which is a Frobenius group with the Frobenius kernel regular on the vertex set, or (iii) Aut M = PSL ( 2 , 2 e ) and M is a non-orientable non-Cayley map.