Using mixed dihedral groups to construct normal Cayley graphs, and a new bipartite $2$-arc-transitive graph which is not a Cayley graph

A \emph{mixed dihedral group} is a group $H$ with two disjoint subgroups $X$ and $Y$, each elementary abelian of order $2^n$, such that $H$ is generated by $X\cup Y$, and $H\mathrm{/}{H}^{\mathrm{\prime }}\cong X\times Y$. In this paper we give a sufficient condition such that the automorphism group of the Cayley graph $\text{Cay}(H,(X\cup Y)\setminus \{1\})$ is equal to $H:A(H,X,Y)$, where $A(H,X,Y)$ is the setwise stabiliser in $\text{Aut}(H)$ of $X\cup Y$. We use this criterion to resolve a questions of Li, Ma and Pan from 2009, by constructing a $2$-arc transitive normal cover of order $2^{53}$ of the complete bipartite graph ${\text{K}}_{16,16}$ and prove that it is \emph{not} a Cayley graph.