Fifth generation (5G) mobile technology enables a new kind of network that provides high peak data rates, ultra-low latency communication and high user density. Electronic healthcare (e-health) allows the data to be stored and shared in a highly efficient and flexible manner. Group e-health services help in improving long-term results of the treatments due to its collaborative characteristic. The services including real-time remote patient monitoring and transmission of large health data files can be facilitated by e-health systems enabled with the 5G network. Since the communication channel is open, security and privacy in the system should be taken into account. Our work proposes a key distribution scheme for group healthcare services in 5G network environments. We construct various healthcare domains and apply the proposed scheme to the group services. The paper also introduces Single Sign-On (SSO), a cost-efficient solution, for multi-server architecture of the constructed system. Security and privacy of the scheme are enforced by a three-factor authentication mechanism (integrating smart card, password and biometrics) and strong user anonymity property. We provide security proof of the proposed scheme using various well-known tools including RoR model, BAN logic and AVISPA simulation. Results of various performance comparisons indicate that our scheme provides most functions and bears rational costs, compared with its related works.