  For all finite fields of $q$ elements where $q\equiv1\pmod4$ we have
constructed permutation polynomials which have order 2 as permutations, and
have 3 terms, or 4 terms as polynomials. Explicit formulas for their
coefficients are given in terms of the primitive elements of the field. We also
give polynomials providing involutions with larger number of terms but
coefficients will be conveniently only two possible values. Our procedure gives
at least $(q-1)/4$ trinomials, and $(q-1)/2$ quadrinomials, all yielding
involutions with unique fixed points over a field of order $q$. Equal number of
involutions with exactly $(q+1)/2$ fixed-points are provided as quadrinomials.
