The classification of $p$-divisible groups over 2-adic discrete valuation rings

Let $\mathscr{O}_K$ be a 2-adic discrete valuation ring with perfect residue field $k$. We classify $p$-divisible groups and $p$-power order finite flat group schemes over $\mathscr{O}_K$ in terms of certain Frobenius module over $\mathfrak{S}:=W(k)[[u]]$. We also show the compatibility with crystalline Dieudonn\'e theory and associated Galois representations. Our approach differs from Lau's generalization of display theory, and we additionally obtain the the compatibility with associated Galois representations.