  In this paper, we prove two results: first, we use crystalline cohomology of
classifying stacks to directly reconstruct the classical Dieudonn\'e module of
a finite, $p$-power rank, commutative group scheme $G$ over a perfect field $k$
of characteristic $p>0$. As a consequence, we give a new proof of the
isomorphism $\sigma^* M(G) \simeq \mathrm{Ext}^1 (G,
\mathcal{O}^{\mathrm{crys}})$ due to Berthelot--Breen--Messing using stacky
methods combined with the theory of de Rham--Witt complexes. Additionally, we
show that finite locally free commutative group schemes of $p$-power rank over
a quasisyntomic base ring embed fully faithfully into the category of prismatic
$F$-gauges, which extends the work of Ansch\"utz and Le Bras on prismatic
Dieudonn\'e theory.
