Normal Cayley graphs and normal Cayley graphs

Let \(G\) be a finite group with identity \(1\) and let \(C \subset G\setminus \{1\}\) be such that \(C^{-1} = C\) and \(gCg^{-1} = C\) for all \(g\in G\). Let \(X = \operatorname{Cay}(G,C)\) be the Cayley graph of \(G\) with connection set \(C\), and let \(K = \operatorname{Aut}{X}\). For any \(g\in G\), we define \(\lambda_g,\rho_g: G\to G\) such that \(x^{\lambda_g} = g^{-1}x\) and \(x^{\rho_g} = gx\), for \(x\in G\). The group \(L_G\) and \(R_G\) are respectively the left-regular and right-regular representations of \(G\). As \(X\) is a Cayley graph, it is clear that \(R_G\leq \operatorname{Aut}{X}\). The subgroup \(L_G\) need not be a subgroup of \(\operatorname{Aut}{X}\), however, in our case since \(C\) is a union of conjugacy classes, \(L_G\leq \operatorname{Aut}{X}\).