  For an odd prime power $q$ satisfying $q\equiv 1\pmod 3$ we construct totally
$2(q-1) $ permutation polyomials, all giving involutory permutations with
exactly $ 1+ \frac{q-1}3$ fixed points. Among them $(q-1)$ polynomials are
trinomials, and the rest are 6-term polynomials.
