Model theory of Galois actions of torsion Abelian groups

We show that the theory of Galois actions of a torsion Abelian group $A$ is companionable if and only if for each prime $p$, the $p$-primary part of $A$ is either finite or it coincides with the Pr"{u}fer $p$-group. We also provide a model-theoretic description of the model companions we obtain.