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This article is about hyperk\"ahler fourfolds $X$ admitting a non-symplecticinvolution $\iota$. The Bloch-Beilinson conjectures predict the way $\iota$should act on certain pieces of the Chow groups of $X$. The main result is averification of this prediction for Fano varieties of lines on certain cubicfourfolds. This has some interesting consequences for the Chow ring of thequotient $X/\iota$.

On the Chow groups of some hyperkaehler fourfolds with a non-symplectic involution II