Recent technological advances in electronics, sensors, and communications devices have facilitated the proliferation of Unmanned Aircraft System (UAS)-aided applications. However, the UAS-aided communications networks are yet to receive sufficient research endeavor. In this paper, we address one of the most important research challenges pertaining to UAS-aided networks comprising adaptive modulation-capable nodes, namely how to fairly maximize the energy efficiency (throughput per energy). For the mobility pattern innate to the UAS, we demonstrate how the adaptive modulation behaves. Furthermore, we formulate the problem as a potential game that is played between the UAS and the network-nodes, and prove its stability, optimality, and convergence. Based upon the potential game, a data collection method is envisioned to maximize the energy efficiency with the fairness constraint. Additionally, we analyze the Price of Anarchy (PoA) of our proposed game. Extensive simulations exhibit the effectiveness of our proposal under varying environments.