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Dermatological diseases pose a major threat to the global health, affecting almost one-third of the world's population. Various studies have demonstrated that early diagnosis and intervention are often critical to prognosis and outcome. To this end, the past decade has witnessed the rapid evolvement of deep learning based smartphone apps, which allow users to conveniently and timely identify issues that have emerged around their skins. In order to collect sufficient data needed by deep learning and at the same time protect patient privacy, federated learning is often used, where individual clients aggregate a global model while keeping datasets local. However, existing federated learning frameworks are mostly designed to optimize the overall performance, while common dermatological datasets are heavily imbalanced. When applying federated learning to such datasets, significant disparities in diagnosis accuracy may occur. To address such a fairness issue, this paper proposes a fairness-aware federated learning framework for dermatological disease diagnosis. The framework is divided into two stages: In the first in-FL stage, clients with different skin types are trained in a federated learning process to construct a global model for all skin types. An automatic weight aggregator is used in this process to assign higher weights to the client with higher loss, and the intensity of the aggregator is determined by the level of difference between losses. In the latter post-FL stage, each client fine-tune its personalized model based on the global model in the in-FL stage. To achieve better fairness, models from different epochs are selected for each client to keep the accuracy difference of different skin types within 0.05. Experiments indicate that our proposed framework effectively improves both fairness and accuracy compared with the state-of-the-art.