On the Galois-invariant part of the Weyl group of the Picard lattice of a K3 surface

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 $k^{\text{s}}$. Let $O(PicX)$ and $O(Pic{X}^{\text{s}})$ denote the group of isometries of the lattices $PicX$ and $Pic{X}^{\text{s}}$, respectively. Let $R_X$ denote the Galois invariant part of the Weyl group of $Pic{X}^{\text{s}}$. One can show that each element in $R_X$ can be restricted to an element of $O(PicX)$. The following question arises: Is the image of the restriction map $R_X\to O(PicX)$ a normal subgroup of $O(PicX)$ for every K3 surface $X$? We show that the answer is negative by giving counterexamples over $k=Q$.