On solvable groups and Cayley graphs

Let φ be Euler's phi function. Let n be a square-free positive integer such that gcd(n,φ(n))=q, q a prime, and if p|n is prime, then q2∤(p−1). We prove that a vertex-transitive graph Γ of order n is isomorphic to a Cayley graph of order n if and only if Aut(Γ) contains a transitive solvable subgroup.