Graphically abelian groups

We define a group G to be graphically abelian if the function g ↦ g − 1 induces an automorphism of every Cayley graph of G . 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 G is graphically abelian if and only if G = E × Q , where E is an elementary abelian 2-group and Q is a quaternion group.