Vision-Language-Action (VLA) models have shown strong potential for generalizable robotic manipulation. During fine-tuning, however, action supervision applies equally across all timesteps, without structured supervision on which manipulation stage the robot is in or what the next gripper-event target should be. This causes failures to concentrate around challenging gripper-event transitions. To address this, we propose StaKe, a plug-in auxiliary supervision framework that automatically derives two complementary signals from demonstration gripper states without manual annotation: a stage classifier that identifies the current manipulation stage, and a keyframe predictor that estimates the target joint action at the next gripper transition. Both are modeled as lightweight auxiliary heads that enrich the learned representations during training, while leaving the base VLA policy architecture and inference loop unchanged. Experiments on bimanual simulation and single-arm Franka real-robot tasks show that StaKe consistently improves success rates (relative gains of 14% and 56%, respectively), with larger improvements on longer-horizon tasks that involve more gripper-event transitions. Ablation studies validate each design choice, and qualitative analysis confirms that the learned representations faithfully track manipulation stages. These results indicate that structured supervision is an effective and general strategy for enhancing VLA fine-tuning in long-horizon manipulation. Project website: https://hi-yuanxu.github.io/StaKe-Web/

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