  We construct the explicit $Q$-operator incorporated with the
$sl_2$-loop-algebra symmetry of the six-vertex model at roots of unity. The
functional relations involving the $Q$-operator, the six-vertex transfer matrix
and fusion matrices are derived from the Bethe equation, parallel to the
Onsager-algebra-symmetry discussion in the superintegrable $N$-state chiral
Potts model. We show that the whole set of functional equations is valid for
the $Q$-operator. Direct calculations in certain cases are also given here for
clearer illustration about the nature of the $Q$-operator in the symmetry study
of root-of-unity six-vertex model from the functional-relation aspect.
