Since the beginning of the century, an increasing amount of air traffic has pushed current aeronautical communication systems to their limits. Therefore, a modernization process is ongoing, envisioning to digitalize previously analog systems and prepare them for future requirements. Among these efforts is the L-Band Digital Aeronautical Communications System (LDACS). It shall replace legacy analog voice communications in the future. Any newly developed system must provide strong cybersecurity, especially when deployed within critical infrastructures. While previous work has been focused on implementing Mutual Authentication and Key Establishment protocols in LDACS, applying security mechanisms in a group-wise fashion has not been evaluated yet. As LDACS control messages apply to all members of an LDACS cell, Group Key Management (GKM) methods are a necessary step in introducing control channel security to LDACS. The objective of this paper is to evaluate GKM procedures to support secure communication within LDACS control channels.