Algebraic cycles on certain hyperkaehler fourfolds with an order 3 non-symplectic automorphism II

Let $X$ be a hyperk\"ahler variety, and assume that $X$ admits a non-symplectic automorphism $\sigma$ of order $k>{1\over 2}\dim X$. 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 cubic fourfolds having an order $3$ non-symplectic automorphism.