Feature selection is demanded in many modern scientific research problems that use high-dimensional data. A typical example is to identify gene signatures that are related to a certain disease from high-dimensional gene expression data. The expression of genes may have grouping structures, for example, a group of co-regulated genes that have similar biological functions tend to have similar expressions. Thus it is preferable to take the grouping structure into consideration to select features. In this paper, we propose a Bayesian Robit regression method with Hyper-LASSO priors (shortened by BayesHL) for feature selection in high dimensional genomic data with grouping structure. The main features of BayesHL include that it discards more aggressively unrelated features than LASSO, and it makes feature selection within groups automatically without a pre-specified grouping structure. We apply BayesHL in gene expression analysis to identify subsets of genes that contribute to the 5-year survival outcome of endometrial cancer (EC) patients. Results show that BayesHL outperforms alternative methods (including LASSO, group LASSO, supervised group LASSO, penalized logistic regression, random forest, neural network, XGBoost and knockoff) in terms of predictive power, sparsity and the ability to uncover grouping structure, and provides insight into the mechanisms of multiple genetic pathways leading to differentiated EC survival outcome.