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Automatic tumor 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 regular CNNs can only exploit translation invariance, ignoring further inherent symmetries existing in medical images such as rotations and reflections. To mitigate this shortcoming, 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 every orientation, which can effectively address the gaps of learning symmetries in existing approaches. Then, to keep segmentation networks globally equivariant, we design distinctive group layers with layerwise symmetry constraints. By exploiting further symmetries, novel segmentation CNNs can dramatically reduce the sample complexity and the redundancy of filters (by roughly 2/3) over regular CNNs. More importantly, based on our novel framework, we show that a newly built GER-UNet outperforms its regular CNN-based counterpart and the state-of-the-art segmentation methods on real-world clinical data. Specifically, the group layers of our segmentation framework can be seamlessly integrated into any popular CNN-based segmentation architectures.