  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.
