  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 $H/H'\cong X\times Y$. In this paper we give a sufficient
condition such that the automorphism group of the Cayley graph $\Cay(H,(X\cup
Y)\setminus\{1\})$ is equal to $H: A(H,X,Y)$, where $A(H,X,Y)$ is the setwise
stabiliser in $\Aut(H)$ 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 $\K_{16,16}$ and
prove that it is \emph{not} a Cayley graph.
