Graphically abelian groups

We define a group GG to be graphically abelian   if the function g↦g−1g↦g−1 induces an automorphism of every Cayley graph of GG. We give equivalent characterizations of graphically abelian groups, note features of the adjacency matrices for Cayley graphs of graphically abelian groups, and show that a non-abelian group GG is graphically abelian if and only if G=E×QG=E×Q, where EE is an elementary abelian 2-group and QQ is a quaternion group.