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The counterpropagating edge states of a two-dimensional topological insulator (TI) carry electrons of opposite spins. We investigate the transport properties of edge states in a two-dimensional TI which is contacted to ferromagnetic leads. The application of a side-gate voltage induces a constriction or quantum point contact (QPC) which couples the two edge channels. The transport properties of the system are calculated via the Keldysh nonequilibrium Green's function method. We found that inter-edge spin-flip coupling can significantly enhance (suppress) the charge current when the magnetization of the leads are anti-parallel (parallel) to one another. On the other hand, spin-conserving inter-edge coupling generally reduces the current by backscattering regardless of the magnetization configuration. The charge current and the conductance as a function of the bias voltage, also exhibit similar trends with respect to spin-flip coupling strength, for both parallel and anti-parallel configurations. Hence, gate voltage modulation of edge states via a QPC can provide a means of modulating the spin or charge current flow in TI-based spintronics devices.