Let $S_n$ be the symmetric group on $n$ elements. The irreducible representations of $S_n$ are parametrised by partitions $\lambda$ of $n$ and are defined already over the integers $\mathbb Z$. Let $\rho^\lambda: S_n \to \operatorname{GL}_m(\mathbb Z)$ be such an irreducible representation. I am curious whether there is a way to characterise when the image of $\rho^{\lambda}$ lies in a proper subgroup of $GL_m$, i.e. $$\rho^\lambda: S_n \to G(\mathbb Z) \subset \operatorname{GL}_m(\mathbb Z),$$ where $G$ is some (algebraic) group, for example the symplectic group $\operatorname{GSp}_{2k}$, when $2k = m$. Ideally, such a characterisation would be given in terms of partitions and Young tableaux.
I'd be grateful for any leads.