  A Helson matrix is an infinite matrix $A = (a_{m,n})_{m,n\geq1}$ such that
the entry $a_{m,n}$ depends only on the product $mn$. We demonstrate that the
orthogonal projection from the Hilbert--Schmidt class $\mathcal{S}_2$ onto the
subspace of Hilbert--Schmidt Helson matrices does not extend to a bounded
operator on the Schatten class $\mathcal{S}_q$ for $1 \leq q \neq 2 < \infty$.
In fact, we prove a more general result showing that a large class of natural
projections onto Helson matrices are unbounded in the $\mathcal{S}_q$-norm for
$1 \leq q \neq 2 < \infty$. Two additional results are also presented.
