A characterisation of edge-affine $2$-arc-transitive covers of ${\text{K}}_{2^n,2^n}$

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 $\mathrm{\Gamma }$ of $H$ such that the clique graph $\mathrm{\Sigma }$ of $\mathrm{\Gamma }$ is a $2$-arc-transitive normal cover of ${\text{K}}_{2^n,2^n}$, with a subgroup of $\text{Aut}(\mathrm{\Sigma })$ inducing a particular \emph{edge-affine} action on ${\text{K}}_{2^n,2^n}$. Conversely, we prove that if $\mathrm{\Sigma }$ is a $2$-arc-transitive normal cover of ${\text{K}}_{2^n,2^n}$, with a subgroup of $\text{Aut}(\mathrm{\Sigma })$ inducing an \emph{edge-affine} action on ${\text{K}}_{2^n,2^n}$, then the line graph $\mathrm{\Gamma }$ of $\mathrm{\Sigma }$ 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 ${\text{K}}_{2^n,2^n}$.