The Solution of a Problem of Godsil on Cubic Cayley Graphs

In this short paper, we give a positive answer to a question of C. D. Godsil (1983,Europ. J. Combin.4, 25–32) regarding automorphisms of cubic Cayley graphs of 2-groups: “If Cay(G, S) is a cubic Cayley graph of a 2-groupGandA=Aut Cay(G, S), doesA1≠1 imply Aut(G, S)≠1?”