We observed dynamical processes of the magnetic moments of photoinjected carriers and resultant perturbed magnetic moment of Mn${}^{2+}$ ions in CdTe/Cd${}_{1\ensuremath{-}x}$Mn${}_{x}$Te quantum wells by means of the femtosecond time-resolved magneto-optical Kerr rotation measurement in transverse magnetic fields. The time-resolved Kerr rotation signal shows that the Larmor precession of Mn${}^{2+}$ moment is initiated by photoinjected carriers. In particular, we observed a time delay in terms of the phase and an amplitude of Mn${}^{2+}$ beats as a function of the external field to confirm the mechanism of perturbing the Mn${}^{2+}$ moment, i.e., a transient $p\ensuremath{-}d$ exchange field, proposed in earlier studies. The motion of the Mn${}^{2+}$ moment is also simulated by the Bloch model including the impulsive transverse exchange fields produced by the conduction electron and heavy-hole spins. From this it is found that the exchange field of the heavy-hole spin contributes dominantly to tipping the Mn${}^{2+}$ moment away from the external field axis. The exchange field estimated by the Bloch model so as to fit experimental data is in reasonable agreement with the exchange field estimated from $p\ensuremath{-}d$ exchange interaction, which also supports the above mechanism.