The rise of machine learning applications at the network edge demands real-time prediction with limited resources, requiring low computational complexity and accurate models. Also, these devices may have poor learning ability due to the lack of samples. Using federated learning enables devices to train machine learning models without sharing their private data. Meta-learning algorithms, particularly few-shot learning, are ideal for federated environments with highly personalized and decentralized training data due to their fast adaptation and good generalization to new tasks. ‎ Despite recent advances, the use of metric-based meta-learning methods remains ambiguous due to ‎their simplicity, as well as their development to improve model learning ability and accuracy in a federated ‎environment. This paper introduces ResFed, a federated meta-learning method specifically tailored for few-shot ‎classification. The approach involves leveraging a pre-trained model and implementing data augmentation within the federated meta-learner, leading to favorable performance outcomes. The experimental results show that our approach, specifically designed for limited data scenarios in the federated environments, significantly improves convergence speed and accuracy. The values of accuracy obtained in CIFAR-100 and Omniglot ‎datasets are 77.44% and 98.24%, respectively. Additionally, when compared to alternative methods, there is a notable reduction in resource costs ranging from 0.4 to 0.61 in diverse scenarios.