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We report the temporal and spectral analysis of three thermonuclear X-ray bursts from 4U 1608-52, observed by the Neutron Star Interior Composition Explorer (NICER) during and just after the outburst observed from the source in 2020. In two of the X-ray bursts, we detect secondary peaks, 30 and 18 seconds after the initial peaks. The secondary peaks show a fast rise exponential decay-like shape resembling a thermonuclear X-ray burst. Time-resolved X-ray spectral analysis reveals that the peak flux, blackbody temperature, and apparent emitting radius values of the initial peaks are in agreement with X-ray bursts previously observed from 4U 1608-52, while the same values for the secondary peaks tend toward the lower end of the distribution of bursts observed from this source. The third X-ray burst, which happened during much lower accretion rates did not show any evidence for a deviation from an exponential decay and was significantly brighter than the previous bursts. We present the properties of the secondary peaks and discuss the events within the framework of short recurrence time bursts or bursts with secondary peaks. We find that the current observations do not fit in standard scenarios and challenge our understanding of flame spreading., Accepted for publication in the Astrophysical Journal