Complete characterization of a class of complete permutation quadrinomials over \(\mathbb{F}_{q^2}\)

Let $q = 2^m$, $Q = 2^k$, and $1 \leq k \leq m-1$. Write $\overline{x} = x^q$. We study complete permutation quadrinomials over $\mathbb{F}_{q^2}$ of the form \[ f(x) = c_0 x^{Q+1} + c_1 x^Q \overline{x} + c_2 x \overline{x}^Q + c_3 \overline{x}^{Q+1},\qquad c_i \in \mathbb{F}_{q^2}. \] When \(k=1\), Tu et al. (Finite Fields Appl. 68: 1-20, 2020) gave a sufficient condition for \(f\) to be a complete permutation polynomial (CPP) over \(\mathbb{F}_{q^2}\). Chan et al. (Finite Fields Appl. 110: 102734, 2026) later proved that this condition is also necessary, and that under this condition \(f\) and \(f+x\) are linearly equivalent to \(x^2\overline{x}\) and \(x^2\overline{x}+γx\), respectively, for some \(γ\in \mathbb{F}_{q^2}^*\) with $\ord(γ^{q-1})=3$. In this paper, we prove that no such CPP exists for \(k>1\), and that the known condition of Chan et al. is complete for \(k=1\). This completes the characterization for all \(Q=2^k\) with $1 \leq k \leq m-1$.