In this paper, we construct the ${\mathbb{Z}}_2^n-$grassmannians by gluing of the ${\mathbb{Z}}_2^n-$domains and give an explicit description of the action of the ${\mathbb{Z}}_2^n-$Lie group $GL(\overrightarrow{m})$ on the ${\mathbb{Z}}_2^n-$grassmannian $G_{\overrightarrow{k}}(\overrightarrow{m})$ in the functor of points language. In particular, we give a concrete proof of the transitively of this action, and the gluing of the local charts of the ${\mathbb{Z}}_2^n-$grassmannian.