  Tutte's 3-flow conjecture asserts that every $4$-edge-connected graph admits
a nowhere-zero $3$-flow. We prove that this conjecture is true for every Cayley
graph of valency at least four on any supersolvable group with a noncyclic
Sylow $2$-subgroup and every Cayley graph of valency at least four on any group
whose derived subgroup is of square-free order.
