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We study the Intra-Halo Stellar Component (IHSC) of Milky Way-mass systems upto galaxy clusters in the Horizon-AGN cosmological hydrodynamical simulation.We identify the IHSC using an improved phase-space galaxy finder algorithmwhich provides an adaptive, physically motivated and shape-independentdefinition of this stellar component, that can be applied to halos of arbitrarymasses. We explore the IHSC mass fraction-total halo's stellar mass,$f_{M*,IHSC}-M*$, relation and the physical drivers of its scatter. We findthat on average the $f_{M*,IHSC}$ increases with $M_{*,tot}$, with the scatterdecreasing strongly with mass from 2 dex at $M_{*,tot}\sim10^{11}M_\odot$ to0.3 dex at group masses. At high masses, $M_{*,tot}>10^{11.5}M_\odot$,$f_{M*,IHSC}$ increases with the number of substructures, and with the massratio between the central galaxy and largest satellite, at fixed $M_{*,tot}$.From mid-size groups and systems below $M_{*,tot}<10^{12}M_\odot$, we find thatthe central galaxy's stellar rotation-to-dispersion velocity ratio, V/{\sigma},displays the strongest (anti)-correlation with $f_{M*,IHSC}$ at fixed$M_{*,tot}$ of all the galaxy and halo properties explored, transitioning from$f_{M*,IHSC}$<0.1% for high V/{\sigma}, to $f_{M*,IHSC}\sim5$% for lowV/{\sigma} galaxies. By studying the $f_{M*,IHSC}$ temporal evolution, we findthat, in the former, mergers not always take place, but if they did, theyhappened early (z>1), while the high $f_{M*,IHSC}$ population displays a muchmore active merger history. In the case of massive groups and galaxy clusters,$M_{*,tot}>10^{12}M_\odot$, a fraction $f_{M*,IHSC}\sim$10-20% is reached at$z\sim1$ and then they evolve across lines of constant $f_{M*,IHSC}$ modulosome small perturbations. Because of the limited simulation's volume, thelatter is only tentative and requires a larger sample of simulated galaxyclusters to confirm.

From Stellar Halos to Intracluster Light: the physics of the Intra-Halo Stellar Component in cosmological hydrodynamical simulations