Permutation polynomials over finite fields providing involutions

The problem of constructing a family of polynomials over a finite field all yielding permutations in a single conjugacy class, namely involutions with specified number of fixed points, is addressed. For an odd prime power q satisfying \(q\equiv 1\pmod 3\) $q\equiv 1(mod3)$ we construct totally \(2(q-1) \) $2(q-1)$ permutation polynomials, all giving involutory permutations with exactly \( 1+ \frac{q-1}{3}\) $1+\frac{q-1}{3}$ fixed points. Among them \((q-1)\) $(q-1)$ polynomials are trinomials, and the rest are 6-term polynomials.