  We use the eROSITA Final Equatorial-Depth Survey (eFEDS) to measure the
rest-frame 0.1-2.4 keV band X-ray luminosities of $\sim$ 600,000 DESI groups
using two different algorithms in the overlap region of the two observations.
These groups span a large redshift range of $0.0 \le z_g \le 1.0$ and group
mass range of $10^{10.76}h^{-1}M_{\odot} \le M_h \le 10^{15.0}h^{-1}M_{\odot}$.
(1) Using the blind detection pipeline of eFEDS, we find that 10932 X-ray
emission peaks can be cross matched with our groups, $\sim 38 \%$ of which have
signal-to-noise ratio $\rm{S}/\rm{N} \geq 3$ in X-ray detection. Comparing to
the numbers reported in previous studies, this matched sample size is a factor
of $\sim 6$ larger. (2) By stacking X-ray maps around groups with similar
masses and redshifts, we measure the average X-ray luminosity of groups as a
function of halo mass in five redshift bins. We find, in a wide halo mass
range, the X-ray luminosity, $L_{\rm X}$, is roughly linearly proportional to
$M_{h}$, and is quite independent to the redshift of the groups. (3) We use a
Poisson distribution to model the X-ray luminosities obtained using two
different algorithms and obtain best-fit $L_{\rm
X}=10^{28.46\pm0.03}M_{h}^{1.024\pm0.002}$ and $L_{\rm X}=10^{26.73 \pm
0.04}M_{h}^{1.140 \pm 0.003}$ scaling relations, respectively. The best-fit
slopes are flatter than the results previously obtained, but closer to a
self-similar prediction.
