In this paper, we propose MOUFLON, a fairness-aware, modularity-based community detection method that allows adjusting the importance of partition quality over fairness outcomes. MOUFLON uses a novel proportional balance fairness metric, providing consistent and comparable fairness scores across multi-group and imbalanced network settings. We evaluate our method under both synthetic and real network datasets, focusing on performance and the trade-off between modularity and fairness in the resulting communities, along with the impact of network characteristics such as size, density, and group distribution. As structural biases can lead to strong alignment between demographic groups and network structure, we also examine scenarios with highly clustered homogeneous groups, to understand how such structures influence fairness outcomes. Our findings showcase the effects of incorporating fairness constraints into modularity-based community detection, and highlight key considerations for designing and benchmarking fairness-aware social network analysis methods.