On a hierarchy of groups of computable automorphisms

A complete description is provided for the groups $\bigoplus\limits_{i\in I} {Z}_{p_i}$ that can be realized as groups of all computable automorphisms for appropriate computable models. We introduce a three-level classification of isomorphism types of groups of computable automorphisms by the possible arithmetical complexity of their orbits and prove this classification to be nontrivial.