The innovation in the automotive domain is shifting considerably to the Electrical/Electronics system and software. The evolution cycle of the electronic system and the software is significantly shorter than the life cycle of a vehicle and therefore the functionality of a vehicle might become 'outdated' easily in the future. Thus, it would be advantageous if new applications can be installed or existing applications can be upgraded via cloud services after sales in a plug-and-play fashion. This requires that the underlying system possesses a certain degree of adaptivity and reconfigurability. One important issue in this case is the allocation of computation and communication resources. In the case of a time-triggered system, the task and network schedules need to be adapted to accommodate new applications. Towards addressing this problem, we propose a schedule management framework to obtain, synthesize and manage schedules efficiently online for Ethernet-based time-triggered systems in the automotive context. This framework is based on a client-server architecture and each side consists of a web module, a synthesis module and a configuration pool. It utilizes the Internet access of modern vehicles to exploit the computation and storage capacity on the server in a cloud-computing manner and can facilitate the reuse of generated schedule sets. In the synthesis module, a four-stage strategy is introduced to reduce the synthesis time and the disturbance to existing applications. The experimental results show that the proposed framework can be applied to generate and manage schedules online and benefit from both onboard and cloud-based schedule synthesis. The result also shows the applicability of the introduced four-stage synthesis~strategy.