  Let $f(X)=X(1+aX^{q(q-1)}+bX^{2(q-1)})\in\Bbb F_{q^2}[X]$, where $a,b\in\Bbb
F_{q^2}^*$. In a series of recent papers by several authors, sufficient
conditions on $a$ and $b$ were found for $f$ to be a permutation polynomial
(PP) of $\Bbb F_{q^2}$ and, in characteristic $2$, the sufficient conditions
were shown to be necessary. In the present paper, we confirm that in
characteristic 3, the sufficient conditions are also necessary. More precisely,
we show that when $\text{char}\,\Bbb F_q=3$, $f$ is a PP of $\Bbb F_{q^2}$ if
and only if $(ab)^q=a(b^{q+1}-a^{q+1})$ and $1-(b/a)^{q+1}$ is a square in
$\Bbb F_q^*$.
