A hierarchical game-based cooperative control scheme is developed for the grouped formation of second-order multi-agent systems under given or externally updated task parameters. First, a three-layer architecture consisting of task-parameter specification, inter-group equilibrium-reference generation, and intra-group coordination is established. The low-dimensional parameter vector $\theta$ performs translation, uniform scaling, and rotation of a prescribed template family, and shapes the unique Nash equilibrium of the leaders through parameterized cost functions. Second, a distributed equilibrium-seeking algorithm that exchanges only aggregative-estimation auxiliary variables is designed for the leaders, so that equilibrium references can be generated without sharing physical positions or velocities. The resulting public communication signals are further shown to be non-uniquely informative about the physical initial states, which yields a communication-layer state-masking effect. For the followers, a control law based on local relative states and trusted intra-group broadcast signals is developed to achieve compact coordination. Furthermore, under constant task parameters, global exponential convergence of the whole closed-loop system to the desired grouped formation is rigorously established, and the steady-state separation between the public auxiliary variables and the actual physical positions is characterized. Finally, a 56-agent example and a piecewise-constant task-switching example illustrate the convergence, template-reconfiguration capability, and communication-layer state-masking property of the proposed method in moderately large-scale scenarios.} \enkeyword{Multi-agent systems; grouped formation; hierarchical games; Nash equilibrium; state masking; cooperative control