Doubly transitive 2-factorizations

Let F be a 2-factorization of the complete graph $K_v$ admitting an automorphism group G acting doubly transitively on the set of vertices. The vertex-set $V(K_v)$ can then be identified with the point-set of AG(n, p) and each 2-factor of F is the union of p-cycles which are obtained from a parallel class of lines of AG(n, p) in a suitable manner, the group G being a subgroup of AGL(n, p) in this case. The proof relies on the classification of 2-(v, k, 1) designs admitting a doubly transitive automorphism group. The same conclusion holds even if G is only assumed to act doubly homogeneously.