The rapid spread of information and rumors through social media platforms, especially in group settings, motivates the need for more sophisticated models of rumor propagation. Traditional pairwise models do not account for group interactions, a limitation that we address by proposing a higher-order rumor model based on hypergraphs. Our model incorporates a group-based annihilation mechanism, where a spreader becomes a stifler when the fraction of hyperedges aware of the rumor exceeds a threshold. Our model has two distinct subcritical behaviors: exponential and power-law decay, which can coexist depending on the heterogeneity of the hypergraph. Interestingly, our analysis reveals continuous phase transitions in both homogeneous and heterogeneous hypergraphs, challenging the idea that higher-order interactions lead to discontinuous transitions. Finally, we validate our model using empirical data from Telegram and email cascades, providing additional evidence that real-world rumor propagation tends to occur near criticality. These results open the door to a more detailed understanding of rumor dynamics in higher-order systems.