Polynomial Dedekind domains with finite residue fields of prime characteristic

We show that every Dedekind domain $R$ lying between the polynomial rings $\Z[X]$ and $\Q[X]$ 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 $p\in\Z$ there exists a finite subset $E_p$ of transcendental elements over $\Q$ in the absolute integral closure $\overline{\Z_p}$ of the ring of $p$-adic integers such that $R=\{f\in\Q[X]\mid f(E_p)\subseteq \overline{\Z_p}, \text{ for each prime }p\in\Z\}$. 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 $\Z[X]$ and $\Q[X]$.