On finite groups all of whose cubic Cayley graphs are integral

For any positive integer k, let G(k) denote the set of finite groups G such that all Cayley graphs Cay(G, S) are integral whenever vertical bar S vertical bar = k. Estelyi and Kovacs [On groups all of whose undirected Cayley graphs of bounded valency are integral, Electron. J. Combin. 21 (2014) #P4.45.] classified Gk for each k = 4. In this paper, we characterize the finite groups each of whose cubic Cayley graphs is integral. Moreover, the class G(3) is characterized. As an application, the classification of Gk is obtained again, where k = 4.