  For a prime $p$, we show that uniqueness of factorization into irreducible
$\Sigma_{p^2}$-invariant representations of $\mathbb{Z}/p \wr \mathbb{Z}/p$
holds if and only if $p=2$. We also show nonuniqueness of factorization for
$\Sigma_8$-invariant representations of $D_8 \wr \mathbb{Z}/2$. The
representation ring of $\Sigma_{p^2}$-invariant representations of
$\mathbb{Z}/p \wr \mathbb{Z}/p$ is determined completely when $p$ equals two or
three.
