With emerging application of Federated Learning (FL) in decision-making scenarios, it is imperative to regulate model fairness to prevent disparities across sensitive groups (e.g., female, male). Current research predominantly focuses on two concepts of group fairness within FL: Global Fairness (overall model disparity across all clients) and Local Fairness (the disparity within each client). However, the non-decomposable, non-differentiable nature of fairness criteria pose two fundamental, unresolved challenges for fair FL: (i) Harmonizing global and local fairness, especially in multi-class classification; (ii) Enabling a controllable, optimal accuracy-fairness trade-off. To tackle the aforementioned challenges, we propose a novel controllable federated group-fairness calibration framework, named FedFACT. FedFACT identifies the Bayes-optimal classifiers under both global and local fairness constraints in multi-class case, yielding models with minimal performance decline while guaranteeing fairness. To effectively realize an adjustable, optimal accuracy-fairness balance, we derive specific characterizations of the Bayes-optimal fair classifiers for reformulating fair FL as personalized cost-sensitive learning problem for in-processing, and bi-level optimization for post-processing. Theoretically, we provide convergence and generalization guarantees for FedFACT to approach the near-optimal accuracy under given fairness levels. Extensive experiments on multiple datasets across various data heterogeneity demonstrate that FedFACT consistently outperforms baselines in balancing accuracy and global-local fairness.