  We study the Intra-Halo Stellar Component (IHSC) of Milky Way-mass systems up
to galaxy clusters in the Horizon-AGN cosmological hydrodynamical simulation.
We identify the IHSC using an improved phase-space galaxy finder algorithm
which provides an adaptive, physically motivated and shape-independent
definition of this stellar component, that can be applied to halos of arbitrary
masses. We explore the IHSC mass fraction-total halo's stellar mass,
$f_{M*,IHSC}-M*$, relation and the physical drivers of its scatter. We find
that on average the $f_{M*,IHSC}$ increases with $M_{*,tot}$, with the scatter
decreasing strongly with mass from 2 dex at $M_{*,tot}\sim10^{11}M_\odot$ to
0.3 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 mass
ratio 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 that
the 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 low
V/{\sigma} galaxies. By studying the $f_{M*,IHSC}$ temporal evolution, we find
that, in the former, mergers not always take place, but if they did, they
happened early (z>1), while the high $f_{M*,IHSC}$ population displays a much
more 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 $f_{M*,IHSC}$ modulo
some small perturbations. Because of the limited simulation's volume, the
latter is only tentative and requires a larger sample of simulated galaxy
clusters to confirm.
