Federated learning (FL) enables training AI models across distributed edge devices (i.e., workers) using local data, while facing challenges including communication resource constraints, edge heterogeneity, and non-IID data. Over-the-air computation (AirComp) has emerged as a promising technique to improve communication efficiency by leveraging the superposition property of a wireless multiple access channel (MAC) for model aggregation. However, over-the-air aggregation requires strict synchronization among edge devices, which is essentially incompatible with the asynchronous FL mechanisms often used to handle edge heterogeneity. To overcome this incompatibility, we propose Air-FedGA, a grouping-based asynchronous FL mechanism via AirComp, where workers are organized into groups for synchronized over-the-air aggregation within each group, while groups asynchronously communicate with the parameter server to update the global model. This design retains the communication efficiency of AirComp while addressing training inefficiency caused by edge heterogeneity. We provide a rigorous convergence analysis for Air-FedGA, theoretically quantifying how the convergence bound depends on several key factors, such as the maximum staleness, the degree of non-IID data among groups, and the AirComp aggregation mean squared error (MSE). Guided by these theoretical insights, we propose power control and worker grouping algorithms to minimize the convergence bound by jointly optimizing the AirComp aggregation MSE and the grouping strategy. We conduct experiments on classical models and datasets, and the results demonstrate that our proposed mechanism and algorithms can accelerate the model training by 1.83-$2.22\times $compared with the state-of-the-art solutions.