On quasiabelian Cayley graphs

Wang and Xu proposed (European J. Combin. 18 (1997) 567–600, Problem 3) to determine those Cayley graphs Γ of a group G satisfying Aut Γ=L(G)R(G), where L(G) and R(G) denote the left and the right regular representations of G. We prove that the equation holds if and only if (1) G≃(Z2)n, where n is a nonnegative integer different from 2,3,4, and (2) Γ is a graphical regular representation of G.