Federated fine-tuning for Large Language Models (LLMs) faces significant challenges due to the heavy communication overhead of transmitting large model updates. Although Low Rank Adaptation (LoRA) has been proposed as a solution, yet its application in federated learning is complicated by discordance in aggregation. Existing methods addressing this discordance often suffer from performance degradation at low ranks in heterogeneous data settings. In response, we introduce LoRA-A$^2$ (Low Rank Adaptation with Alternating freeze and Adaptive rank selection), which demonstrates robustness in challenging settings with low ranks and high data heterogeneity. Our experimental findings reveal that LoRA-A$^2$ maintains performance even under extreme heterogeneity and low rank conditions, achieving up to a significant reduction in uploaded parameters compared to full fine-tuning without compromising performance. This adaptive mechanism increases robustness and communication efficiency in federated fine-tuning, enabling the practical deployment of LLMs in resource-constrained environments.