### SCYON Abstract

We calculate the stellar integrated galactic initial mass function (IGIMF) in the presence of cluster-to-cluster variations of the IMF. Variations of the IMF for a population of coeval clusters that populate the initial cluster mass function (ICLMF) are taken into account in the form of Gaussian distribution functions of the IMF parameters. For the tapered power-law function used in this work, these are the slope at the high-mass end, $\Gamma$, the slope at the low-mass end, $\gamma$, and the characteristic mass $M_{ch}$. The level of variations is modeled by varying the width of the Gaussian distributions. The reference values are the standard deviations of the parameters observed for the population of young clusters in the present-day Milky Way, which are $\sigma_{\Gamma}=0.6$, $\sigma_{\gamma}=0.25$, and $\sigma_{M_{ch}}=0.27$ M$_{\odot}$. We find that increasing the levels of dispersion for $\gamma$ and $\Gamma$ tends to moderately flatten the IGIMF at the low and high-mass end, respectively. The characteristic mass of the IGIMF is, however, strongly impacted by variations in $M_{ch}$. Increasing the value of $\sigma_{M_{ch}}$ shifts the peak of the IGIMF to lower masses, rendering the IGIMF more bottom heavy. This can provide a simple explanation for the bottom-heavy stellar mass function that is inferred for early-type galaxies since these are likely the result of a merger of disk galaxies where the physical conditions of the star-forming gas may vary significantly both in time and space in the merging system. The effect of IMF variations on the IGIMF is compared to the effects of other processes and sources of systematic variations such as those due to variations in the shape of ICLMF, the gas-phase metallicity, and the galactic star formation rate (SFR) which can potentially affect the maximum mass of stellar clusters in a galaxy and set the mean value of the characteristic mass in clusters. For the various dependencies we have explored, we found that the effect of IMF variations is a dominant factor that always affects the characteristic mass of the IGIMF. For the regimes at low metallicity where the IGIMF resembles a single power law, an increased level of IMF variations renders the IGIMF steeper and more bottom heavy, especially at low SFRs. On the other hand, variations in the IMF in the high mass regime can be easily dominated by variations in the slope of the ICLMF. We compare our results of the metallicity and SFR-dependent IGIMF to a sample of Milky Way ultra-faint dwarf (UFD) satellite galaxies that have available metallicity measurements. The present-day stellar mass function of these galaxies is a good analog to the IGIMF at the time their overall population of stars formed. We show that the slope of the stellar mass function of the UFD galaxies measured for stars in the mass range [0.4,0.8] M$_{\odot}$ can only be reproduced when IMF variations of the same order as those measured in the present-day Milky Way are included. Our results suggest that the inclusion of IMF variations in models of galaxy formation and evolution is of vital importance in order to improve our understanding of star formation and star formation feedback effects on galactic scales.