Federated Learning (FL) has garnered significant interest recently due to its potential as an effective solution for tackling many challenges in diverse application scenarios, for example, data privacy in network edge traffic classification. Despite its recognized advantages, FL encounters obstacles linked to statistical data heterogeneity and labeled data scarcity during the training of single-task models for machine learning-based traffic classification, leading to hindered learning performance. In response to these challenges, adopting a hard-parameter sharing multi-task learning model with auxiliary tasks proves to be a suitable approach. Such a model has the capability to reduce communication and computation costs, navigate statistical complexities inherent in FL contexts, and overcome labeled data scarcity by leveraging knowledge derived from interconnected auxiliary tasks. This paper introduces a new framework for federated auxiliary hard-parameter sharing multi-task learning, namely, FedAuxHMTL. The introduced framework incorporates model parameter exchanges between edge server and base stations, enabling base stations from distributed areas to participate in the FedAuxHMTL process and enhance the learning performance of the main task-network edge traffic classification. Empirical experiments are conducted to validate and demonstrate the FedAuxHMTL's effectiveness in terms of accuracy, total global loss, communication costs, computing time, and energy consumption compared to its counterparts.