  The $N$-state chiral Potts model in lattice statistical mechanics can be
obtained as a ``descendant'' of the six-vertex model, via an intermediate
``$Q$'' or ``$\tau_2 (t_q)$'' model. Here we generalize this to obtain a
column-inhomogeneous $\tau_2 (t_q)$ model, and derive the functional relations
satisfied by its row-to-row transfer matrix. We do {\em not} need the usual
chiral Potts relations between the $N$th powers of the rapidity parameters
$a_p, b_p, c_p, d_p$ of each column. This enables us to readily consider the
case of fixed-spin boundary conditions on the left and right-most columns. We
thereby re-derive the simple direct product structure of the transfer matrix
eigenvalues of this model, which is closely related to the superintegrable
chiral Potts model with fixed-spin boundary conditions.
