Autonomous vehicles are a growing technology that aims to enhance safety, accessibility, efficiency, and convenience through autonomous maneuvers ranging from lane change to overtaking. Overtaking is one of the most challenging maneuvers for autonomous vehicles, and current techniques for autonomous overtaking are limited to simple situations. This paper studies how to increase safety in autonomous overtaking by allowing the maneuver to be aborted. We propose a decision-making process based on a deep Q-Network to determine if and when the overtaking maneuver needs to be aborted. The proposed algorithm is empirically evaluated in simulation with varying traffic situations, indicating that the proposed method improves safety during overtaking maneuvers. Furthermore, the approach is demonstrated in real-world experiments using the autonomous shuttle iseAuto.