  Let $X$ be a hyperk\"ahler variety, and let $G$ be a group of finite order
non-symplectic automorphisms of $X$. Beauville's conjectural splitting property
predicts that each Chow group of $X$ should split in a finite number of pieces.
The Bloch-Beilinson conjectures predict how $G$ should act on these pieces of
the Chow groups: certain pieces should be invariant under $G$, while certain
other pieces should not contain any non-trivial $G$-invariant cycle. We can
prove this for two pieces of the Chow groups when $X$ is the Hilbert scheme of
a $K3$ surface and $G$ consists of natural automorphisms. This has consequences
for the Chow ring of the quotient $X/G$.
