On a class of complete permutation quadrinomials

Let $f\left(x\right)=a{x}^{3q}+b{x}^{2q+1}+c{x}^{q+2}+d{x}^3\in F_{q^2}\left[x\right]$, where $F_{q^2}$ is the finite field of order $q^2$ and $q=2^m$ for some positive integer m. Tu et al. (Finite Fields Appl. 68: 1-20, 2020) proposed a sufficient condition under which $f\left(x\right)$ is a complete permutation on $F_{q^2}$. In this paper, we show that this sufficient condition is also necessary, and when $f\left(x\right)$ is a complete permutation, then $f\left(x\right)$ and $f\left(x\right)+x$ are simultaneously linear equivalent to $x^2\bar{x}$ and $x^2\bar{x}+\gamma x$ for some $\gamma \in F_{q^2}^{⁎}$ satisfying $\mathrm{ord}\left({\gamma }^{q-1}\right)=3$. This result leads to a complete characterization of the complete permutation quadrinomials of the above form $f\left(x\right)$.