Machine learning models for medical image analysis often exhibit subgroup-dependent performance, which impacts how decisions should be allocated between automated systems and human experts under limited resources. Prior work on AI fairness and human-AI cooperation, including learning to defer (L2D) and learning to complement (L2C), typically addresses these problems in isolation. We propose People-Centred Medical Image Analysis (PecMan), a framework for fairness-aware human-AI co-operative classification that jointly models subgroup-dependent reliability, decision allocation, and collaborative prediction. PecMan combines subgroup-specialised predictors with a gating and consolidation mechanism that dynamically assigns cases to automated models, human experts, or their combination, without requiring sensitive attributes at test time. We also introduce the FairHAI benchmark for evaluating trade-offs between predictive accuracy, subgroup equity, and human involvement. In addition, we provide a theoretical analysis of multi-agent gating via selection regret and characterise fairness-coverage trade-offs under input-dependent allocation. Experiments across multiple medical imaging datasets demonstrate that PecMan achieves consistently improved trade-offs compared to methods that address fairness or human-AI cooperation separately.