  Let $f={\tt X}^r(a+{\tt X}^{2(q-1)})\in{\Bbb F}_{q^2}[{\tt X}]$, where
$a\in{\Bbb F}_{q^2}^*$ and $r\ge 1$. The parameters $(q,r,a)$ for which $f$ is
a permutation polynomial (PP) of ${\Bbb F}_{q^2}$ have been determined in the
following cases: (i) $a^{q+1}=1$; (ii) $r=1$; (iii) $r=3$. These parameters
together form three infinite families. For $r>3$ (there is a good reason not to
consider $r=2$) and $a^{q+1}\ne 1$, computer search suggested that $f$ is not a
PP of ${\Bbb F}_{q^2}$ when $q$ is not too small relative to $r$. In the
present paper, we prove that this claim is true. In particular, for each $r>3$,
there are only finitely many $(q,a)$, where $a^{q+1}\ne 1$, for which $f$ is a
PP of ${\Bbb F}_{q^2}$.
