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

Determination of a Class of Permutation Trinomials in Characteristic Three