We estimate the average group morphological and dynamical characteristics of the Percolation-Inferred Galaxy Group (2PIGG) catalog within z ≤ 0.08, for which the group space density is roughly constant. We quantify the different biases that enter in the determination of these characteristics, and we devise statistical correction procedures to recover their bias-free values. We find that the only acceptable morphological model is that of prolate, or triaxial with pronounced prolateness, group shapes having a roughly Gaussian intrinsic axial ratio distribution with mean ~0.46 and dispersion ~0.16. After correcting for various biases, the most important of which is a redshift-dependent bias, the median values of the virial mass and virial radius of groups with 4-30 galaxy members are v ~ 6 × 1012 h M☉, v ~ 0.4 h Mpc, which are significantly smaller than recent literature values that do not take into account the previously mentioned biases. The group mean crossing time is ~1.5 Gyr, independent of the group galaxy membership. We also find that there is a correlation of the group size, velocity dispersion, and virial mass with the number of group member galaxies, a manifestation of the hierarchy of cosmic structures.