  This note concerns hyperk\"ahler fourfolds $X$ having a non-symplectic
involution $\iota$. The Bloch-Beilinson conjectures predict the way $\iota$
should act on certain pieces of the Chow groups of $X$. The main result is a
verification of this prediction for Fano varieties of lines on certain cubic
fourfolds. This has consequences for the Chow ring of the quotient $X/\iota$.
