The task scheduling of energy harvesting embedded systems (EHES) should take into account the energy supply of energy harvesting unit, the energy level of energy storage unit and the energy consumption of energy dissipation unit. A real-time task can meet time constraint only if its energy constraint is satisfied. Against this background, conventional fixed-priority tasks scheduling algorithms are not suitable for EHES. A group-based adaptive task scheduling algorithm is proposed in this paper. It can select suitable task scheduling algorithm adaptively according to the non-energy constraint condition and the energy constraint condition caused by the uncertain energy supply of energy harvesting unit. In the case of non-energy constraints, the algorithm can reduce the tasks preemptions and enhance the tasks schedulability. In the case of energy constraints, the algorithm can reduce the battery-mode switches and increase the average energy level of energy storage unit, thus decrease the system energy constraint. The proposed algorithm is validated with large scale simulations comparing with other two existing classical algorithms.