Fusion-invariant representations for symmetric groups

For a prime $p$, we show that uniqueness of factorization into irreducible ${\mathrm{\Sigma }}_{p^2}$-invariant representations of $\mathbb{Z}\mathrm{/}p\wr \mathbb{Z}\mathrm{/}p$ holds if and only if $p=2$. We also show nonuniqueness of factorization for ${\mathrm{\Sigma }}_8$-invariant representations of $D_8\wr \mathbb{Z}\mathrm{/}2$. The representation ring of ${\mathrm{\Sigma }}_{p^2}$-invariant representations of $\mathbb{Z}\mathrm{/}p\wr \mathbb{Z}\mathrm{/}p$ is determined completely when $p$ equals two or three.