This work is devoted to a theoretical analysis of the effect of nuclear-induced field (Overhauser field) on the Larmor frequencies of electron spins under the periodic pulsed excitation. To describe the dynamical nuclear spin polarization, we use the model where the optically induced Stark field determines the magnitude and direction of the Overhauser field. The Stark field strongly depends on the detuning between the photon energy of excitation and the optical transition energy in the quantum system. Detailed calculations which show that the precession frequencies of fluorine donor-bound electron spins in ZnSe deviate from the linear dependence of the Larmor frequencies on the external magnetic field have been performed. A similar effect is observed for the (In,Ga)As/GaAs quantum dots, where it has been shown that the Overhauser field strongly changes the spectrum of the electron spin precession frequencies.