Deploying federated learning (FL) algorithms in wireless edge networks presents significant challenges, including data heterogeneity at individual and group levels, and user availability. Current research primarily focuses on addressing individual-level data heterogeneity with various personalized FL (PFL) algorithms, while challenges persist in handling group-level heterogeneity and understanding the impact of user availability on the PFL process. To bridge this research gap, we propose an availability-aware group-personalized FL algorithm for generic wireless edge networks. Our algorithm effectively detects user availability during the learning process and addresses group-level data heterogeneity. Through rigorous analysis, we demonstrate its convergence performance under suitable conditions. Simulation results showcase the algorithm’s robustness in handling fluctuations in user availability and its effectiveness in addressing group-level data heterogeneity. Our work highlights the practical implications and potential applications of the proposed algorithm in real-world scenarios.