The solution of a problem of Godsil on cubic Cayley graphs (Q1386425)

For a finite group \(G\) and a subset \(S\) of \(G\setminus\{1\}\) with \(S= S^{-1}: = \{s^{-1} \mid s\in S\}\), the associated Cayley graph \(\Gamma: =\text{Cay} (G,S)\) is the graph with vertex set \(V\Gamma =G\) such that \(\{x,y\}\) is an edge if and only if \(yx^{-1}\in S\). Let \(A\) be the automorphism group of \(\Gamma\), and let \(A_1\) be the stabilizer of 1 in \(A\). Set \(\Aut(G,S) =\{\alpha \in\Aut (G)\mid S^\alpha =S\}\). If \(A_1=1\) then \(\Aut (G,S)=1\), but the converse is not necessarily ture. \textit{C. D. Godsil} [Eur. J. Comb. 4, 25-32 (1983; Zbl 0507.05038)] asked if \(G\) is a 2-group and \(\Gamma\) is a connected Cayley graph of \(G\) of valency 3, does \(A_1\neq 1\) imply \(\Aut (G,S)\neq 1\)? This paper gives a positive answer to this question, and thus for a 2-group \(G\) and a connected cubic Cayley graph \(\Gamma\) of \(G\), \(A_1=1\) if and only if \(\Aut (G,S)=1\).