$\newcommand\dual{^{\text{dual}}}\newcommand\GrpSch{\mathrm{GrpSch}}\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\Vect{Vect}$If $G$ is a finite group scheme over a field $k$, we can define its Cartier dual $G\dual$ as the group scheme representing the functor
$$G\dual(B) := \Hom_{\GrpSch_B}(G_{B}, (\mathbb{G}_{m})_{B}),$$
the group scheme homomorphisms between the base-changes. One can then show that there is a canonical isomorphism of vector spaces
$$\mathcal{O}_{G\dual} = \Hom_{\Vect(k)}(\mathcal{O}_{G}, k).$$
In fact, this is an isomorphism of Hopf algebras, so that classical Cartier duality of finite group schemes is the same as the linear duality in the setting of finite Hopf algebras.
Now suppose that $A$ is a ring of integers in a $p$-adic field with residue field $k$. In this context, Faltings defines (see Faltings - Group schemes with strict $\mathcal O$-action) a Cartier duality for finite $k$-scheme $A$-modules (with some strictness conditions which I will ignore here). Concretely, if $G$ is a strict $A$-module scheme, then $G^{\text{$A$-dual}}$ is defined as the representing object for
$$G^{\text{$A$-dual}}(B) := \Hom_{A\text-\GrpSch_B}(G_{B}, (\mathbb{G}^\text{LT})_{B}),$$
where $\mathbb{G}^\text{LT}$ is the Lubin–Tate formal group of $A$.
Question: Is it possible to explicitly describe the coordinate ring $\mathcal{O}_{G^{\text{$A$-dual}}}$ of the $A$-Cartier dual in terms of the coordinate ring of $G$? How is Faltings $A$-Cartier duality related to linear duality of Hopf algebras, or Cartier duality of underlying group schemes?
