Recently incentive mechanisms in crowdsensing systems have received considerable attention for persuading participation in sensing activities. However, designing an incentive mechanism for group activities sensing tasks is still challenging. Furthermore, developing an incentive mechanism that is able to reward the participants according to their weighted contribution over stream sensing tasks is not well considered. To address these issues in this article we propose a Group Reliability-aware Incentive Mechanism (G-RIM) over mobile crowdsensing data streams. G-RIM is a weight-based incentive mechanism that rewards the most contributed group as well as their contributed members in a continuous sensing task. The basic idea is that the most contributed group is selected based on the measures of the weight of the sensing data provided by the group and their members in each time slot, which is estimated by the truth discovery process. The theoretical proofs demonstrate that G-RIM achieves computational efficiency, individual rationality, budget feasibility, truthfulness, and strategy-proof properties. We have conducted extensive experiments over synthetic and two real-world datasets to prove the effectiveness and efficiency of our incentive mechanism. The results show that G-RIM outperforms the Benchmark scheme and the current weight-based incentive scheme in terms of average incentive reward with a reasonable reward distribution time.