Let be a connected tetravalent Cayley graph on a regular p-group G and let be the automorphism group of G. In this paper, it is proved that, for each prime , the automorphism group of the Cayley graph is the semidirect product where is the right regular representation of G and . The proof depends on the classification of finite simple groups. This implies that if then the Cayley graph is normal, namely, the automorphism group of contains as a normal subgroup.