  In a companion article (referred hearafter 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 specialising our considerations to
the case of vacuum and $\gamma -$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 \emph{general
}self-similar solution for the exceptional type VI$_{-1/9}$ model and we
identify it as equilibrium point in the corresponding dynamical state space. It
is found that this \emph{new} exact solution belongs to the subclass of models
$n_\alpha ^\alpha =0$, is defined for $\gamma \in (\frac 43,\frac 32)$ and
although has a five dimensional stable manifold there exist always 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.
