Automatic tumor or lesion segmentation is a crucial step in medical image analysis for computer-aided diagnosis. Although the existing methods based on convolutional neural networks (CNNs) have achieved the state-of-the-art performance, many challenges still remain in medical tumor segmentation. This is because, although the human visual system can detect symmetries in 2-D images effectively, regular CNNs can only exploit translation invariance, overlooking further inherent symmetries existing in medical images, such as rotations and reflections. To solve this problem, we propose a novel group equivariant segmentation framework by encoding those inherent symmetries for learning more precise representations. First, kernel-based equivariant operations are devised on each orientation, which allows it to effectively address the gaps of learning symmetries in existing approaches. Then, to keep segmentation networks globally equivariant, we design distinctive group layers with layer-wise symmetry constraints. Finally, based on our novel framework, extensive experiments conducted on real-world clinical data demonstrate that a group equivariant Res-UNet (called GER-UNet) outperforms its regular CNN-based counterpart and the state-of-the-art segmentation methods in the tasks of hepatic tumor segmentation, COVID-19 lung infection segmentation, and retinal vessel detection. More importantly, the newly built GER-UNet also shows potential in reducing the sample complexity and the redundancy of filters, upgrading current segmentation CNNs, and delineating organs on other medical imaging modalities.