  We aim at characterizing the hot phase of the Circum-Galactic Medium in a
large sample of galaxies. We stack X-ray events from the SRG/eROSITA eFEDS
survey around central galaxies in the GAMA 9hr field to construct radially
projected soft X-ray luminosity profiles as a function of their stellar mass
and specific star formation rate. We consider samples of quiescent
(star-forming) galaxies in the stellar mass range $2\times 10^{10}$ --
$10^{12}$ M$_\odot$ ($3\times 10^9$ -- $6\times10^{11}$ M$_\odot$). For
quiescent galaxies, the X-ray profiles are clearly extended throughout the
available mass range; however, the measured profile is likely biased high due
to projection effects, as these galaxies tend to live in dense and hot
environments. For the most massive star forming samples ($\geq10^{11}$
M$_\odot$), there is a hint of detection of extended emission. For star-forming
galaxies with $< 10^{11}$ M$_\odot$ the X-ray stacked profiles are compatible
with unresolved sources and consistent with the expected emission from faint
Active Galactic Nuclei and X-ray binaries. We measure for the first time the
mean relation between average X-ray luminosity and stellar mass separately for
quiescent and star-forming galaxies. High-mass ($\geq 10^{11}$ M$_\odot$)
star-forming or quiescent galaxies follow the expected scaling of virialized
hot haloes, while lower mass star-forming galaxies show a less prominent
luminosity and a weaker dependence on stellar mass, consistent with empirical
models of the weak AGN population. When comparing our results with state-of-the
art numerical simulations, we find an overall consistency on large ($>80$ kpc)
scales at masses $\geq 10^{11}$ M$_\odot$, but disagreement on the small
scales, where brighter than observed compact cores are predicted. Simulations
also do not predict the clear differentiation that we observe between quiescent
and star-forming galaxies.
