For a general cubic fourfold $Y$ with associated Fano variety of lines $ F $, we show that the monodromy group of the finite degree 16 rational Voisin self-map $ψ\colon F \dashrightarrow F$ is maximal. To achieve this, we investigate the intriguing interplay between $ ψ$ and the fixed locus of the antisymplectic involution on the LLSvS variety $ Z $, examined via the degree 6 Voisin map $F \times F \dashrightarrow Z $.

v1:18 pages, Macaulay2 code included as an ancillary file. v2: Minor changes. v3:final version to appear on EPIGA. Comments are welcome!