The use of telemedicine is arguably beneficial even in densely populated areas in reducing cost and increasing efficiency of healthcare. However, the implementation of telemedicine solutions in the healthcare system of Denmark has been perceived as being faced with implementation and interoperability issues, silo solutions, and lack of guidelines and standards. In this paper, we characterise the ecosystem evolved around the telemedicine services in Denmark and study the actors involved in this ecosystem. We establish a method for this study, where we define two actor roles and ways of characterizing actor contributions, and apply the method to the largest healthcare region of Denmark. Our findings reveal an ecosystem that is relatively closed to new actors, where the actors tend to be related to single telemedicine applications, the applications have low connectivity, and the most influential actors of the ecosystem can be characterised as both being beneficial and inhibitory to the ecosystem prosperity.