We present mass and entropy radial profiles for a sample of 16 relaxed X-ray bright galaxy groups-poor clusters (kT range 1-3 keV, M_vir range 10^13 - 2 x 10^14 solar masses) selected for optimal mass constraints from the Chandra and XMM archives. After accounting for the mass of the hot gas, the resulting mass profiles are described well by a two-component model consisting of dark matter represented by an NFW model, and stars form the central galaxy. We find for the first time that the NFW concentration parameter (c_vir) for groups decreases with increasing M_vir. The normalization and slope of the c_vir - m_vir relation are consistent with the standard LambdaCDM cosmological model with sigma_8 = 0.9. The mean gas fraction (f_gas = 0.05 +/- 0.01) of the groups measured within an over-density of 2500 is lower than for hot, massive cluster with a larger fractional scatter (sigma_f_gas/f_gas = 0.2) implying a greater impact of feedback processes on groups. This is reflected in the entropy profiles which show striking deviations from self-similarity, large object-to-object variation and a characteristic broken-power-law behavior, unlike more massive clusters. We also present the interesting case of the core of one object in the sample, the relaxed poor cluster AWM 4 which is characterized by a unique combination of properties which defy the newly established paradigm for AGN heating in cluster cores: a flat inner temperature profile indicative of a past, major heating episode which completely erased the cool core, as testified by the high central cooling time (> 3 Gyrs) and by the high central entropy level (~ 60 keV cm^2), and yet an active central radio galaxy with extended radio lobes out to 100 kpc, revealing recent feeding of the central massive black hole.