Exceptional 2-to-1 rational functions

For each odd prime power q, we describe a class of rational functions $f\left(X\right)\in F_q\left(X\right)$ with the following unusual property: for every odd j, the function induced by $f\left(X\right)$ on $F_{q^j}\cup \{\infty \}$ is 2-to-1. We also show that, among all known rational functions $f\left(X\right)\in F_q\left(X\right)$ which are 2-to-1 on $F_{q^j}\cup \{\infty \}$ for infinitely many j, our new functions are the only ones which cannot be written as compositions of rational functions of degree at most four, monomials, Dickson polynomials, and additive (linearized) polynomials.