  Let $X$ denote a K3 surface over an arbitrary field $k$. Let $k^\text{s}$
denote a separable closure of $k$ and let $X^\text{s}$ denote the base change
of $X$ to $k^\text{s}$. The action of the absolute Galois group
Gal($k^\text{s}/k$) of $k$ on Pic $X^\text{s}$ respects the intersection
pairing, which gives Pic $X^\text{s}$ the structure of a lattice. Let O(Pic
$X$) and O(Pic $X^\text{s}$) denote the group of isometries of Pic $X$ and Pic
$X^\text{s}$, respectively. Let $R_X$ denote the Galois invariant part of the
Weyl group of O(Pic $X^\text{s}$). One can show that each element in $R_X$ can
be restricted to an element of O(Pic $X$). The following question arises: Is
the image of the restriction map $R_X \to $O(Pic $X$) a normal subgroup of
O(Pic $X$) for every K3 surface $X$? We show that the answer is negative by
giving counterexamples over $k=\mathbb{Q}$.
