Complete mappings stabilizing Sylow $2$-subgroups in central quotients of special linear groups

A complete mapping of a finite group $G$ is a permutation $θ$ of $G$ such that $x\mapsto xθ(x)$ is also a permutation of $G$. Complete mappings have applications in cryptography, particularly in the design of round functions for cryptographic systems. The study of their parity has led to the questio