  The problem of finding upper bounds for minimal vertex number of graphs with
a given automorphism group is addressed in this article for the case of cyclic
$2$-groups. We show that for any natural $n\ge 2$ there is an undirected graph
having $2^n+6$ vertices and automorphism group cyclic of order $2^n$. This
confirms an upper bound claimed by other authors for minimal number of vertices
of undirected graphs having automorphism group $\mathbb{Z}/2^n\mathbb{Z}$.
