Automated and accurate EGFR mutation status prediction using computed tomography (CT) imagery is of great value for tailoring optimal treatments to non-small cell lung cancer (NSCLC) patients. However, existing deep learning based methods usually adopt a single task learning strategy to design and train EGFR mutation status prediction models with limited training data, which may be insufficient to learn distinguishable representations for promoting prediction performance. In this paper, a novel multi-task learning method named AIR-Net is proposed to precisely predict EGFR mutation status on CT images. First, an auxiliary image reconstruction task is effectively integrated with EGFR mutation status prediction, aiming at providing extra supervision at the training phase. Particularly, we adequately employ multi-level information in a shared encoder to generate more comprehensive representations of tumors. Second, a powerful feature consistency loss is further introduced to constrain semantic consistency of original and reconstructed images, which contributes to enhanced image reconstruction and offers more effective regularization to AIR-Net during training. Performance analysis of AIR-Net indicates that auxiliary image reconstruction plays an essential role in identifying EGFR mutation status. Furthermore, extensive experimental results demonstrate that our method achieves favorable performance against other competitive prediction methods. All the results executed in this study suggest that the effectiveness and superiority of AIR-Net in precisely predicting EGFR mutation status of NSCLC.