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GPUs can provide excellent computational performance at a good price point for tasks that can be suitably parallelized. Charged particle (track) reconstruction is a computationally expensive component of HEP data reconstruction, and thus needs to use available resources in an efficient way. In this paper, an implementation of Kalman filter-based track fitting using CUDA and running on GPUs is presented. This utilizes the ACTS (A Common Tracking Software) toolkit; an open source and experiment-independent toolkit for track reconstruction. The implementation details and parallelization approach are described, along with the specific challenges for such an implementation. Detailed performance benchmarking results are discussed, which show encouraging performance gains over a CPU-based implementation for representative configurations. Finally, a perspective on the challenges and future directions for these studies is outlined. 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