Let ${𝒪}_K$ be a discrete valuation ring of mixed characteristics $(0,p)$, with residue field $k$. Using work of Sekiguchi and Suwa, we construct some finite flat ${𝒪}_K$-models of the group scheme ${\mu }_{p^n,K}$ of $p^n$-th roots of unity, which we call Kummer group schemes. We carefully set out the general framework and algebraic properties of this construction. When $k$ is perfect and ${𝒪}_K$ is a complete totally ramified extension of the ring of Witt vectors $W\left(k\right)$, we provide a parallel study of the Breuil-Kisin modules of finite flat models...