Properties of Breuil-Kisin modules inherited by $p$-divisible groups

In this paper, by assuming a faithful action of a finite flat ${\mathbb{Z}}_p$-algebra $\mathcal{R}$ on a $p$-divisible group $\mathcal{G}$ defined over the ring of $p$-adic integers ${\mathcal{O}}_K$, we construct a category of new Breuil-Kisin module $\mathfrak{M}$ defined over the ring $\mathfrak{S}:=W(\kappa )[\text{\hspace{0.17em}⁣}[u]\text{\hspace{0.17em}⁣}]$ and study the freeness and projectiveness properties of such a module.