Half-arc-transitive graphs of order 4p of valency twice a prime

A graph is half-arc-transitive if its automorphism group acts transitively on vertices and edges, but not on arcs. Let p be a prime. Cheng and Oxley [On weakly symmetric graphs of order twice a prime, J. Combin. Theory B 42(1987) 196-211] proved that there is no half-arc-transitive graph of order 2p, and Alspach and Xu [1/2-transitive graphs of order 3p, J. Algebraic Combin. 3(1994) 347-355] classified half-arc-transitive graphs of order 3p. In this paper we classify half-arc-transitive graphs of order 4p of valency 2q for each prime q >= 5. It is shown that such graphs exist if and only if p - 1 is divisible by 4q. Moreover, for such p and q a unique half-arc-transitive graph of order 4p and valency 2q exists and this graph is a Cayley graph.