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Interfaces between ferromagnetic metals and nonmagnetic specimen attract much attention as they are very important for the formation of magnetic properties of nanostructures. Vice versa, specific magnetic ordering at such interfaces may provide new effects in their optical and nonlinear optical response. In this work we study the magnetization-induced effects in optical second harmonic generation (SHG) in W/Co/Pt-based thin films with the thicknesses of the Co layer of 2-10 nm. Besides common "odd" in magnetization effects in the SHG intensity, we observe additional one that is not expected for homogeneously magnetized ferromagnetic films, which consists in modulation of p-polarized SHG intensity under longitudinal magnetic field application. The phenomenological description of the observed effect is performed in terms of gradient and second-order in magnetization contributions to the SHG polarization, where gradient of magnetization along the normal to the structure plays the key role.