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Asynchronous tasks, when created with over-decomposition, enable automatic computation-communication overlap which can substantially improve performance and scalability. This is not only applicable to traditional CPU-based systems, but also to modern GPU-accelerated platforms. While the ability to hide communication behind computation can be highly effective in weak scaling scenarios, performance begins to suffer with smaller problem sizes or in strong scaling due to fine-grained overheads and reduced room for overlap. In this work, we integrate GPU-aware communication into asynchronous tasks in addition to computation-communication overlap, with the goal of reducing time spent in communication and further increasing GPU utilization. We demonstrate the performance impact of our approach using a proxy application that performs the Jacobi iterative method, Jacobi3D. In addition to optimizations to minimize synchronizations between the host and GPU devices and increase the concurrency of GPU operations, we explore techniques such as kernel fusion and CUDA Graphs to mitigate fine-grained overheads at scale.