  Let $X$ be a hyperk\"ahler variety, and assume $X$ has a non-symplectic
automorphism $\sigma$ of order $>{1\over 2}\dim X$. Bloch's conjecture predicts
that the quotient $X/<\sigma>$ should have trivial Chow group of $0$-cycles. We
verify this for Fano varieties of lines on certain special cubic fourfolds
having an order $3$ non--symplectic automorphism.
