  We aim to investigate the bolometric $L_{\mathrm{X}}-T$ relation for galaxy
groups, and study the impact of gas cooling, feedback from supermassive black
holes, and selection effects on it. With a sample of 26 galaxy groups we
obtained the best fit $L_{\mathrm{X}}-T$ relation for five different cases
depending on the ICM core properties and central AGN radio emission, and
determined the slopes, normalisations, intrinsic and statistical scatters for
both temperature and luminosity. Simulations were undertaken to correct for
selection effects (e.g. Malmquist bias) and the bias corrected relations for
groups and clusters were compared. The slope of the bias corrected
$L_{\mathrm{X}}-T$ relation is marginally steeper but consistent with clusters
($\sim 3$). Groups with a central cooling time less than 1 Gyr (SCC groups)
show indications of having the steepest slope and the highest normalisation.
For the groups, the bias corrected intrinsic scatter in $L_{\mathrm{X}}$ is
larger than the observed scatter for most cases, which is reported here for the
first time. Lastly, we see indications that the groups with an extended central
radio source have a much steeper slope than those groups which have a CRS with
only core emission. Additionally, we also see indications that the more
powerful radio AGN are preferentially located in NSCC groups rather than SCC
groups.
