Aims. We aim to investigate the bolometric L X − T relation for galaxy groups, and to study the impact of gas cooling, feedback from super-massive black holes, and the selection effects on it.Methods. With a sample of 26 galaxy groups, we obtained the best-fit L X − T relation for five different cases depending on the intracluster medium (ICM) core properties and central active galactic nuclei (AGN) radio emission, and determined the slopes, normalisations, and intrinsic and statistical scatters for both temperature and luminosity. We undertook simulations to correct for selection effects (e.g. Malmquist bias) and compared the bias-corrected relations for groups and clusters. Results. The slope of the bias-corrected L X − T relation is marginally steeper, but consistent with clusters (∼3). Groups with a central cooling time of 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 X is larger than the observed scatter for most cases, and this is reported here for the first time. We see indications that the groups with an extended central radio source (CRS) have a much steeper slope than those groups that have a CRS with only core emission. Additionally, we see indications that the more powerful radio AGN are preferentially located in non strong cool core (NSCC) groups rather than strong cool core (SCC) groups.