Just a few things that come to my mind; most are about the base ring $\mathbb{Z}$, but a good answer would tell us something interesting about $\mathbb{Q}$ as well.

1. Specht modules can be defined not just for partitions and for skew partitions, but for any shape, i.e., any finite set of "boxes" in $\mathbb{Z}^2$. For example, you can define it as the span of the polytabloids in the module of tabloids, or you can (equivalently) define it as the left ideal of the symmetric group algebra generated by the Young symmetrizer $b_T a_T$, where $T$ is any tableau of the given shape, $a_T$ is its row-symmetrizer and $b_T$ is its column-antisymmetrizer. For details, see §1.2 of Ricky Ini Liu, Specht modules and Schubert varieties for general diagrams, thesis 2010. How does such a general Specht module decompose into irreducibles? Even more basically, what is a basis of this module? Can its dimension depend on the characteristic of the base field? Various particular cases have been studied, such as the %-avoiding shapes (Reiner/Shimozono 1998).

2. The Gelfand-Zetlin subalgebra of the group algebra $\mathbb Z\left[S_n\right]$ is the subring (= $\mathbb Z$-subalgebra) generated by the Young-Jucys-Murphy elements $m_k := t_{1,k} + t_{2,k} + \cdots + t_{k-1,k}$, where $t_{i,j}$ is the transposition swapping $i$ with $j$. What is a basis of this subalgebra as a $\mathbb{Z}$-module? Is it a saturated $\mathbb{Z}$-submodule (i.e., a direct summand of $\mathbb Z\left[S_n\right]$ as a $\mathbb{Z}$-module)? Note that over $\mathbb{Q}$, it has a well-known basis, namely the diagonal part of the seminormal basis of $\mathbb Q\left[S_n\right]$ (that is, the family $\left(e_{T,T}\right)$ with $T$ ranging over the standard tableaux with $n$ cells).

Just a few things that come to my mind; most are about the base ring $\mathbb{Z}$, but a good answer would tell us something interesting about $\mathbb{Q}$ as well.

1. Specht modules can be defined not just for partitions and for skew partitions, but for any shape, i.e., any finite set of "boxes" in $\mathbb{Z}^2$. For example, you can define it as the span of the polytabloids in the module of tabloids, or you can (equivalently) define it as the left ideal of the symmetric group algebra generated by the Young symmetrizer $b_T a_T$, where $T$ is any tableau of the given shape, $a_T$ is its row-symmetrizer and $b_T$ is its column-antisymmetrizer. For details, see §1.2 of Ricky Ini Liu, Specht modules and Schubert varieties for general diagrams, thesis 2010. How does such a general Specht module decompose into irreducibles? Even more basically, what is a basis of this module? Can its dimension depend on the characteristic of the base field? Various particular cases have been studied, such as the %-avoiding shapes (Reiner/Shimozono 1998).

2. The Gelfand-Zetlin subalgebra of the group algebra $\mathbb Z\left[S_n\right]$ is the subring (= $\mathbb Z$-subalgebra) generated by the Young-Jucys-Murphy elements $m_k := t_{1,k} + t_{2,k} + \cdots + t_{k-1,k}$, where $t_{i,j}$ is the transposition swapping $i$ with $j$. What is a basis of this subalgebra as a $\mathbb{Z}$-module? Is it a saturated $\mathbb{Z}$-submodule (i.e., a direct summand of $\mathbb Z\left[S_n\right]$ as a $\mathbb{Z}$-module)? Note that over $\mathbb{Q}$, it has a well-known basis, namely the diagonal part of the seminormal basis of $\mathbb Q\left[S_n\right]$ (that is, the family $\left(e_{T,T}\right)$ with $T$ ranging over the standard tableaux with $n$ cells).

EDIT: See https://www.cip.ifi.lmu.de/~grinberg/algebra/nfe2024.pdf for more details on these problems (and a third one on the group ring of $S_n$, albeit not representation-theoretical).