Let $\Gamma=Cay(G,S)$ be a connected Cayley (di)graph over a group of order twice a prime and $\Sigma$ be a complete system of 2-blocks for $Aut(\Gamma)$. Let $K$ be the kernel of the action of $Aut(\Gamma)$ on $\Sigma$ and $K'$ be the kernel of the action of $R(G)$ on $\Sigma$. I need to prove that when $K'$ is a proper subgroup of $K$, then there is a subgroup of $H\leq G$ of order 2 that $S-H$ is union of some double cosets of $H$.
Thanks for every one who trid to answer the question. I have an answer for my question. Let $B\in\Sigma$. Consider the action of $K$ on $B$. If this action is faithful, then $K=S_2$. On the other hand $S_2= K'< K$. So the action of $K$ on $B$ is unfaithful. So by Lemma 2.1 of "On the Normality of Cayley Graphs of order $pq$, Z.P. Lu and M.Y.Xu, Australian Journal of Combinatorics, 27(2003) 81-93", $\Gamma$ is a lexicographic product. Now by Theorem 2.2 of "Cayley digraphs and lexicographic product, P. Xing and W.Dianjun,Front. Math. China 2007, 2(3): 1-8" the result is clear.