Given an action of a discrete group G on a smooth compact manifold M with a boundary, we consider a class of operators generated by pseudodifferential operators on M and shift operators associated with the group action. For elliptic operators in this class, we obtain a classification up to stable homotopies and show that the group of stable homotopy classes of such problems is isomorphic to the K-group of the crossed product of the algebra of continuous functions on the cotangent bundle over the interior of the manifold and the group G acting on this algebra by automorphisms. © Savin A. Yu., Sternin B. Yu. 2016.