We present the experimental studies on optical second-harmonic generation from nanostructures exhibiting magnetic dipole resonances in the visible spectral range. "Nanosandwiches" of the composition Au/MgF(2)/Au with the average disk diameter of 140 nm are packed in a square lattice with the period of 400 nm. We show that at normal incidence, the intensity of the second-harmonic (SH) wave generated by the array of nanostructures increases by an order of magnitude as the excitation wavelength approaches the magnetic-dipole resonance, while the phase of the SH wave experiences a shift up to 330°. Based on the phenomenological description of the SH process, the observed effects indicate the dominant role of the nonlinear magnetic-dipole polarization driven by the χ(emm) susceptibility in SH generation in "nanosandwiches."