AGN heating, through massive subrelativistic outflows, might be the key to solve the long-lasting `cooling flow problem' in cosmological systems. In a previous paper, we showed that cold accretion feedback and, to a lesser degree, Bondi self-regulated models are in fact able to quench cooling rates for several Gyr, at the same time preserving the mainc ool core features, like observed density and temperature profiles. Is it true also for lighter systems, such as galaxy groups? The answer is globally yes, although with remarkable differences. Adopting a modified version of the AMR code FLASH 3.2, we found that successful 3D simulations with cold and Bondi models are almost convergent in the galaxy group environment, with mechanical efficiencies in the range 5.e-4 - 1.e-3 and 5.e-2 - 1.e-1, respectively. The evolutionary storyline of galaxy groups is dominated by a quasi-continuous gentle injection with sub-Eddington outflows (with mechanical power and velocity around 1.e44 erg/s and 1.e4 km/s). The cold and hybrid accretion models present, in addition, very short quiescence periods, followed by moderate outbursts (10 times the previous phase), which generate a series of 10-20 kpc size cavities with high density contrast, temperatures similar to the ambient medium and cold rims. After shock heating, a phase of turbulence promotes gas mixing and diffusion of metals, which peak along jet-axis (up to 40 kpc) during active phases. At this stage the tunnel, produced by the enduring outflow (hard to detect in the mock SBx maps), is easily fragmented, producing tiny buoyant bubbles, typically a few kpc in size. In contrast to galaxy clusters, the AGN self-regulated feedback has to be persistent, with a `delicate touch', rather than rare and explosive strokes. This evolutionary difference dictates in the end that galaxy groups are not scaled-down versions of clusters. ; Comment: Accepted by MNRAS; 22 pages, 7 figures