A generalization of the map $\chi$

The mapping ${\chi }_n:{\mathbb{F}}_2^n\to {\mathbb{F}}_2^n$ defined by $y={\chi }_n(x)$ with $y_i=x_i+x_{i+1}{x}_{i+2}+x_{i+2}$, where the indices are computed modulo $n$, has been widely studied for its application in lightweight cryptography. In this paper, we generalize this mapping and completely characterize all these shift-invariant permutations of algebraic degree $2$.