Polynomial Dedekind domains with finite residue fields of prime characteristic

We show that every Dedekind domain $R$ lying between the polynomial rings $\mathbb{Z}[X]$ and $\mathbb{Q}[X]$ with the property that its residue fields of prime characteristic are finite fields is equal to a generalized ring of integer-valued polynomials, that is, for each prime $p\in \mathbb{Z}$ there exists a finite subset $E_p$ of transcendental elements over $\mathbb{Q}$ in the absolute integral closure $\overline{{\mathbb{Z}}_p}$ of the ring of $p$-adic integers such that $R=\{f\in \mathbb{Q}[X]\mid f(E_p)\subseteq \overline{{\mathbb{Z}}_p},\mathrm{\forall }\text{ prime }p\in \mathbb{Z}\}$. Moreover, we prove that the class group of $R$ is isomorphic to a direct sum of a countable family of finitely generated abelian groups. Conversely, any group of this kind is the class group of a Dedekind domain $R$ between $\mathbb{Z}[X]$ and $\mathbb{Q}[X]$.