While significant progress has been made in conventional fairness-aware machine learning (ML) and differentially private ML (DPML), the fairness of privacy protection across groups remains underexplored. Existing studies have proposed methods to assess group privacy risks, but these are based on the average-case privacy risks of data records. Such approaches may underestimate the group privacy risks, thereby potentially underestimating the disparity across group privacy risks. Moreover, the current method for assessing the worst-case privacy risks of data records is time-consuming, limiting their practical applicability. To address these limitations, we introduce a novel membership inference game that can efficiently audit the approximate worst-case privacy risks of data records. Experimental results demonstrate that our method provides a more stringent measurement of group privacy risks, yielding a reliable assessment of the disparity in group privacy risks. Furthermore, to promote privacy protection fairness in DPML, we enhance the standard DP-SGD algorithm with an adaptive group-specific gradient clipping strategy, inspired by the design of canaries in differential privacy auditing studies. Extensive experiments confirm that our algorithm effectively reduces the disparity in group privacy risks, thereby enhancing the fairness of privacy protection in DPML.