Automorphism groups of tetravalent Cayley graphs on regular p-groups

Let Cay(G,S) be a connected tetravalent Cayley graph on a regular p-group G and let Aut(G) be the automorphism group of G. In this paper, it is proved that, for each prime p≠2,5, the automorphism group of the Cayley graph Cay(G,S) is the semidirect product R(G)⋊Aut(G,S) where R(G) is the right regular representation of G and Aut(G,S)={α∈Aut(G)|Sα=S}. The proof depends on the classification of finite simple groups. This implies that if p≠2,5 then the Cayley graph Cay(G,S) is normal, namely, the automorphism group of Cay(G,S) contains R(G) as a normal subgroup.