The ability to switch ferroics (magnets, ferroelectrics, multiferroics) between two stable bit states is the main principle of modern data storage technology. Due to many new ideas, originating from fundamental research during the last 50 years, this technology has developed in a breath-taking fashion. Ever increasing demands for faster and more energy efficient data storage strongly motivate fundamental studies of dynamics in ferroics triggered by ultrashort stimuli. It has been recently realized that nearly single cycle intense THz pulses and the phenomenon of the second harmonic generation are appealing tools for excitation and detection of poorly understood ultrafast dynamics of electric polarization in ferroelectrics at the picosecond timescale. Here we investigate picosecond dynamics of second harmonic from near-infrared pulse in ferroelectric heterostructure Ba0.8Sr0.2TiO3/MgO triggered by the electric field of a nearly single cycle intense THz pulse. The dynamics of the nonlinear optical signal is characterized by a step and oscillations at the frequency of about 1.67 THz. Although the observations can be mistakenly interpreted as oscillations of the electric polarization at the frequency of the soft mode and switching of the order parameter to another metastable state, here we show that the THz modulation of second harmonic generation in Ba0.8Sr0.2TiO3/MgO has a purely optical origin. The observation can be explained assuming that the THz pulse is a relativistically propagating inhomogeneity which induces center of symmetry breaking and linear birefringence. Our work reveals the role of propagation effects in interpretation of time-resolved non-linear optical experiments and thus it has important implications for experimental studies of ultrafast dynamics in ferroics.