Group robustness has become a major concern in machine learning (ML) as conventional training paradigms were found to produce high error on minority groups. Without explicit group annotations, proposed solutions rely on heuristics that aim to identify and then amplify the minority samples during training. In our work, we first uncover a critical shortcoming of these methods: an inability to distinguish legitimate minority samples from poison samples in the training set. By amplifying poison samples as well, group robustness methods inadvertently boost the success rate of an adversary---e.g., from 0\% without amplification to over 97\% with it. Notably, we supplement our empirical evidence with an impossibility result proving this inability of a standard heuristic under some assumptions. Moreover, scrutinizing recent poisoning defenses both in centralized and federated learning, we observe that they rely on similar heuristics to identify which samples should be eliminated as poisons. In consequence, minority samples are eliminated along with poisons, which damages group robustness---e.g., from 55\% without the removal of the minority samples to 41\% with it. Finally, as they pursue opposing goals using similar heuristics, our attempt to alleviate the trade-off by combining group robustness methods and poisoning defenses falls short. By exposing this tension, we also hope to highlight how benchmark-driven ML scholarship can obscure the trade-offs among different metrics with potentially detrimental consequences.