Infinite families of APN permutations in constrained trivariate classes over ${\mathbb{F}}_{2^m}$

We study trivariate permutation polynomials over ${\mathbb{F}}_{2^m}$ extending two APN permutation families of Li–Kaleyski (IEEE Trans. Inform. Theory, 2024) by allowing the scalar parameter to vary over ${\mathbb{F}}_{2^m}^{\ast }$. For $$G_a(x,y,z)=(x^{q+1}+a{x}^{q}z+y{z}^q,x^{q}z+y^{q+1},x{y}^q+a{y}^{q}z+z^{q+1}),$$ where $a\in {\mathbb{F}}_{2^m}^{\ast }$, $q=2^i$, $\gcd (i,m)=1$, and $m$ is odd, we prove that $G_a$ is a permutation if and only if an associated univariate polynomial has no root in ${\mathbb{F}}_{2^m}^{\ast }$, and that this condition is also equivalent to $G_a$ being APN. Hence, writing $d=q^2+q+1$, at least $$\frac{2^m+1-(d-1)(d-2)2^{m\mathrm{/}2}-d}{d}$$ values of $a$ yield APN permutations $G_a$. In the binary case $q=2$, we show that $a=1$ is good whenever $7\nmid m$, recovering the Li–Kaleyski family. For the second family $$H_a(x,y,z)=(x^{q+1}+ax{y}^q+y{z}^q,x{y}^q+z^{q+1},x^{q}z+y^{q+1}+a{y}^{q}z),$$ we obtain the same root criterion and prove that its defining polynomial is root-equivalent to that of $G_a$. Thus the same parameters $a$ give APN permutations in both families. We also prove strong inequivalence results. First, $G_a$ (resp.\ $H_a$) is diagonally equivalent to $G_1$ (resp.\ $H_1$) if and only if $a^{q^2+q+1}=1$; moreover, for $m>4$, $m\mathrm{\ne }6$, and $7\nmid m$, diagonal non-equivalence implies CCZ non-equivalence by the monomial restriction theorem of Shi et al.\ (DCC, 2025). In particular, when $q=2$ and $7\nmid m$, every good $a\mathrm{\ne }1$ gives APN permutations CCZ-inequivalent to Li–Kaleyski. Second, for the same range of $m$, no $G_a$ is CCZ-equivalent to any $H_b$. Hence these constructions yield two genuinely new, mutually inequivalent families of APN permutations on ${\mathbb{F}}_{2^{3m}}$.