We investigate the impact of a galaxy's merger history on its system of satellites using the new VINTERGATAN-GM suite of zoom-in hydrodynamical simulations of Milky Way-mass systems. The suite simulates five realizations of the same halo with targeted 'genetic modifications' of a z ≈ 2 merger, but resulting in the same halo mass at z = 0. We find that differences in the satellite stellar mass functions last for 2.25-4.25 Gyr after the z ≈ 2 merger; specifically, the haloes that have undergone smaller mergers host up to 60 per cent more satellites than those of the larger merger scenarios. However, by z = 0 these differences in the satellite stellar mass functions have been erased. The differences in satellite numbers seen soon after the mergers are driven by several factors, including the timings of significant mergers (with M_200c mass ratios >1:30 and bringing in M_* ≥ 10⁸ M_⊙ at infall), the masses and satellite populations of the central and merging systems, and the subsequent extended history of smaller mergers. The results persist when measured at fixed central stellar mass rather than fixed time, implying that a host's recent merger history can be a significant source of scatter when reconstructing its dynamical properties from its satellite population.