Isomorphism between Cayley (di)graphs

Let image and image be finite groups. We give a sufficient condition to prove that every Cayley graph of image is isomorphic to a Cayley graph of image. As an application of this result, it is proved that every Cayley graph of a certain group of order 12 is isomorphic to a Cayley graph of the dihedral group of order 12. Analogously, it is proved that every Cayley graph of a cyclic group of order image is isomorphic to a Cayley graph of the dihedral group image, and the converse holds if and only if image. For Cayley digraphs it is proved that every Cayley digraph of image, generated with image, is isomorphic to a Cayley digraph in image.