A 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, for each $n\geq 2$, we construct a mixed dihedral $2$-group $H$ of nilpotency class $3$ and order $2^a$ where $a=(n^3+n^2+4n)/2$, and a corresponding graph $Σ$, which is the clique graph of a Cayley graph of $H$. We prove that $Σ$ is semisymmetric, that is, ${\rm Aut}(Σ)$ acts transitively on the edges, but intransitively on the vertices, of $Σ$. These graphs are the first known semisymmetric graphs constructed from groups that are not $2$-generated (indeed $H$ requires $2n$ generators). Additionally, we prove that $Σ$ is locally $2$-arc-transitive, and is a normal cover of the `basic' locally $2$-arc-transitive graph ${\rm K}_{2^n,2^n}$. As such, the construction of this family of graphs contributes to the investigation of normal covers of prime-power order of basic locally $2$-arc-transitive graphs -- the `local' analogue of a question posed by C.~H.~Li.