  This note contains some examples of hyperk\"ahler varieties $X$ having a
group $G$ of non-symplectic automorphisms, and such that the action of $G$ on
certain Chow groups of $X$ is as predicted by Bloch's conjecture. The examples
range in dimension from $6$ to $132$. For each example, the quotient $Y=X/G$ is
a Calabi-Yau variety which has interesting Chow-theoretic properties; in
particular, the variety $Y$ satisfies (part of) a strong version of the
Beauville-Voisin conjecture.
