Isomorphism between Cayley (di)graphs

Let Γ and Δ be finite groups. We give a sufficient condition to prove that every Cayley graph of Γ is isomorphic to a Cayley graph of Δ. 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 2k is isomorphic to a Cayley graph of the dihedral group Dk, and the converse holds if and only if k∈{2,3,5}. For Cayley digraphs it is proved that every Cayley digraph of Z2k, generated with H⊆{2α}α=1k-1, is isomorphic to a Cayley digraph in Dk.