In a companion article (referred to hereafter as paper I) a detailed study of the simply transitive spatially homogeneous (SH) models of class A concerning the existence of a simply transitive similarity group has been given. The present work (paper II) continues and completes the above study by considering the remaining set of class B models. Following the procedure of paper I we find all SH models of class B subjected only to the minimal geometric assumption to admit a proper homothetic vector field (HVF). The physical implications of the obtained geometric results are studied by specializing our considerations to the case of vacuum and γ-law perfect fluid models. As a result, we regain all the known exact solutions regarding vacuum and non-tilted perfect fluid models. In the case of tilted fluids, we find the general self-similar solution for the exceptional type VI_−1/9 model and we identify it as an equilibrium point in the corresponding dynamical state space. It is found that this new exact solution belongs to the subclass of models n^α_α = 0, is defined for , and although it has a five-dimensional stable manifold there always exist two unstable modes in the restricted state space. Furthermore, the analysis of the remaining types guarantees that tilted perfect fluid models of types III, IV, V and VII_h cannot admit a proper HVF, strongly suggesting that these models either may not be asymptotically self-similar (type V) or may be extreme tilted at late times. Finally, for each Bianchi type, we give the extreme tilted equilibrium points of their state space.