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Chiral molecules are known to preferentially select electrons with a particular spin state, an effect termed chirality-induced spin selectivity (CISS). In this work, the transient CISS dynamics in a chiral molecule are investigated through time-dependent quantum-transport simulations, an important step toward further understanding CISS and its application in devices such as magnetoresistive random access memories and spin-based quantum computers. We show that the spin-dependent group velocity of electrons is a possible contributor to a nonzero occupancy-based spin polarization throughout the chiral molecule. Contrary to the case in which a chiral molecule is connected to a single lead, this spin polarization persists into the steady state when two leads are connected. We show that the simulated spin polarization qualitatively agrees with a reference experiment, as evidenced by the distinct magnetic-field signatures calculated from the spin polarization within a monolayer of chiral molecules.