  We introduce the notion of an \emph{$n$-dimensional mixed dihedral group}, a
general class of groups for which we give a graph theoretic characterisation.
In particular, if $H$ is an $n$-dimensional mixed dihedral group then the we
construct an edge-transitive Cayley graph $\Gamma$ of $H$ such that the clique
graph $\Sigma$ of $\Gamma$ is a $2$-arc-transitive normal cover of
$\K_{2^n,2^n}$, with a subgroup of $\Aut(\Sigma)$ inducing a particular
\emph{edge-affine} action on $\K_{2^n,2^n}$. Conversely, we prove that if
$\Sigma$ is a $2$-arc-transitive normal cover of $\K_{2^n,2^n}$, with a
subgroup of $\Aut(\Sigma)$ inducing an \emph{edge-affine} action on
$\K_{2^n,2^n}$, then the line graph $\Gamma$ of $\Sigma$ is a Cayley graph of
an $n$-dimensional mixed dihedral group. Furthermore, we give an explicit
construction of a family of $n$-dimensional mixed dihedral groups. This family
addresses a problem proposed by Li concerning normal covers of prime power
order of the `basic' $2$-arc-transitive graphs. In particular, we construct,
for each $n\geq 2$, a $2$-arc-transitive normal cover of $2$-power order of the
`basic' graph $\K_{2^n,2^n}$.
