Complete Walsh spectra for a permutation-inverse family of Boolean functions

Let $q=2^e$ with $e$ even, and let $\mathbb{F}_{q^2}$ be the finite field of order $q^2$. Put $d=(q^2+q+1)/3$, and consider the permutation polynomial $$\sigma(X)=X+X^d+X^{dq}\in\mathbb{F}_{q^2}[X].$$ For $\alpha\in\mathbb{F}_q^*$, define the Boolean function $$f_{\alpha}(x)=\text{Tr}_{q^2}\bigl(\alpha(\sigma^{-1}(x))^3\bigr),\qquad x\in\mathbb{F}_{q^2},$$ where $\text{Tr}_{q^2}$ denotes the absolute trace from $\mathbb{F}_{q^2}$ to $\mathbb{F}_{2}$. In this paper, we determine all Walsh values of $f_{\alpha}$ and their multiplicities. In particular, $f_{\alpha}$ is bent if and only if $\alpha$ is a noncube in $\mathbb{F}_q$, proving a conjecture of Li, Li, Helleseth, and Qu. The $\beta\in\mathbb{F}_q$ part of the spectrum is handled by an elementary finite-field argument, whereas the $\beta\in\mathbb{F}_{q^2}\setminus\mathbb{F}_q$ part is reduced to a Hadamard problem on the trace-zero space. We then identify the resulting word with a normalized odd-dimensional shortened Kerdock shell in the noncube case and with a normalized first shortened Delsarte--Goethals shell in the cube case. To make the external shell input independently checkable, we separate the finite-field dictionary from the cited shell theorem and verify directly the degenerate case $e=2$. As an application, we show that a recent cyclotomic family of Xie, Li, Wang, and Zeng is the same construction in different coordinates.