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Binary Neural Networks (BNNs), known to be one among the effectively compact network architectures, have achieved great outcomes in the visual tasks. Designing efficient binary architectures is not trivial due to the binary nature of the network. In this paper, we propose a use of evolutionary search to facilitate the construction and training scheme when binarizing MobileNet, a compact network with separable depth-wise convolution. Inspired by one-shot architecture search frameworks, we manipulate the idea of group convolution to design efficient 1-Bit Convolutional Neural Networks (CNNs), assuming an approximately optimal trade-off between computational cost and model accuracy. Our objective is to come up with a tiny yet efficient binary neural architecture by exploring the best candidates of the group convolution while optimizing the model performance in terms of complexity and latency. The approach is threefold. First, we train strong baseline binary networks with a wide range of random group combinations at each convolutional layer. This set-up gives the binary neural networks a capability of preserving essential information through layers. Second, to find a good set of hyperparameters for group convolutions we make use of the evolutionary search which leverages the exploration of efficient 1-bit models. Lastly, these binary models are trained from scratch in a usual manner to achieve the final binary model. Various experiments on ImageNet are conducted to show that following our construction guideline, the final model achieves 60.09% Top-1 accuracy and outperforms the state-of-the-art CI-BCNN with the same computational cost.