<p>In this study, we examine the stability conditions for set optimization problems under perturbations in both feasible sets and objective maps. We explore convergence conditions for two primary types of efficient solutions of these problems, including weakly efficient and efficient solutions. For weakly efficient solutions, we utilize conditions of generalized Hausdorff convergence, Hausdorff cone-continuity of maps, and relaxed notions of closedness and compactness of sets to establish both external and internal stability. For efficient solutions, these conditions are complemented by the introduction of the induction phenomenon property and the domination property to address external and internal stability. Furthermore, we investigate the existence of such solutions for reference problems even when the compactness of feasible sets is not satisfied. Several examples are presented to illustrate the validity of the assumptions used in our obtained results.</p>