Image classification models trained on clean data often suffer from significant performance degradation when exposed to testing corrupted data, such as images with impulse noise, Gaussian noise, or environmental noise. This degradation not only impacts overall performance but also disproportionately affects various demographic subgroups, raising critical algorithmic bias concerns. Although robust learning algorithms like Sharpness-Aware Minimization (SAM) have shown promise in improving overall model robustness and generalization, they fall short in addressing the biased performance degradation across demographic subgroups. Existing fairness-aware machine learning methods - such as fairness constraints and reweighing strategies - aim to reduce performance disparities but hardly maintain robust and equitable accuracy across demographic subgroups when faced with data corruption. This reveals an inherent tension between robustness and fairness when dealing with corrupted data. To address these challenges, we introduce one novel metric specifically designed to assess performance degradation across subgroups under data corruption. Additionally, we propose \textbf{FairSAM}, a new framework that integrates \underline{Fair}ness-oriented strategies into \underline{SAM} to deliver equalized performance across demographic groups under corrupted conditions. Our experiments on multiple real-world datasets and various predictive tasks show that FairSAM successfully reconciles robustness and fairness, offering a structured solution for equitable and resilient image classification in the presence of data corruption.