We consider semi-supervised binary classification for applications in which data points are naturally grouped (e.g., survey responses grouped by state) and the labeled data is biased (e.g., survey respondents are not representative of the population). The groups overlap in the feature space and consequently the input-output patterns are related across the groups. To model the inherent structure in such data, we assume the partition-projected class-conditional invariance across groups, defined in terms of the group-agnostic feature space. We demonstrate that under this assumption, the group carries additional information about the class, over the group-agnostic features, with provably improved area under the ROC curve. Further assuming invariance of partition-projected class-conditional distributions across both labeled and unlabeled data, we derive a semi-supervised algorithm that explicitly leverages the structure to learn an optimal, group-aware, probability-calibrated classifier, despite the bias in the labeled data. Experiments on synthetic and real data demonstrate the efficacy of our algorithm over suitable baselines and ablative models, spanning standard supervised and semi-supervised learning approaches, with and without incorporating the group directly as a feature.