Determination of a type of permutation binomials over finite fields

Let f=ax+x3q−2∈Fq2[x], where a∈Fq2⁎. We prove that f is a permutation polynomial of Fq2 if and only if one of the following occurs: (i) q=2e, e odd, and aq+13 is a primitive 3rd root of unity. (ii) (q,a) belongs to a finite set which is determined in the paper.