Minimum Size of a Poset Realizing ${\mathbb{Z}}_2\times {\mathbb{Z}}_{2^n}$ as its Automorphism Group

We study the realization of finite groups as automorphism groups of finite posets. Given a finite group $G$, let $\beta (G)$ denote the smallest number of elements in a poset $P$ with $\text{Aut}(P)\cong G$. While $\beta (G)$ is known for several cyclic and small abelian groups, the non-cyclic abelian case is largely open. In this paper we prove that $\beta ({\mathbb{Z}}_2\times {\mathbb{Z}}_{2^n})=2^{n+1}+2$ for every $n\ge 3$.