This article is about hyperkähler fourfolds X admitting a non-symplectic involution ι. The Bloch-Beilinson conjectures predict the way ι 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 some interesting consequences for the Chow ring of the quotient X/ι.(X) should only depend on the subring H * ,0 (X), we arrive at the following conjecture: