Young's rule states that
The Specht module $S^{\alpha}$ appears in the decomposition of $\mathbb Q [S_n/H_{\beta}]$ into irreducible $\mathbb Q [S_n]$-modules precisely $K_{\alpha,\beta}$ times.
Here, $\alpha$ and $\beta$ are partitions of the same integer $n$, $\beta = (\beta_1, \cdots, \beta_l)$, $K_{\alpha,\beta}$ is the Kostka number and $S_{\beta}$ a subgroup of $S_n$ of the form $S_{\beta_1} \times \cdots \times S_{\beta_l}$. (See Theorem 1 or [2, 2.8.5].) In other words,
The multiplicity of the Specht module $S^{\alpha}$ in the induced module $\operatorname{Ind}^{S_n}_{H_{\beta}}(1)$ is $K_{\alpha,\beta}$.
My question is
Are there any results on the multiplicity of $S_{\alpha}$ in $\operatorname{Ind}^{S_n}_H$ where $H$ is not of the form $S_{\beta_1} \times \cdots \times S_{\beta_l}$ for a partition $(\beta_1, \cdots, \beta_l)$?
For example, let $n = 3r$ for $r \in \mathbb Z_+$. Let $$G = \{ \sigma \in S_n | \sigma(i) - \sigma(j) = i - j \text{ for } i,j \in \{kr+1,kr+2, \cdots, (k+1)r \}, k \in \{0,1,2 \} \}.$$ (Obviously, $G \simeq S_3$.) Let $H$ be the subgroup of $S_n$ generated by $S_r \times S_r \times S_r$ and $G$. Then
what is the muliplicity of $S^{\alpha}$ in $\operatorname{Ind}^{S_n}_H(1)$ for any partition $\alpha$ of $n$?
Thank you very much for any answer, comment or hint!
[2]: Gordon James and Adalbert Kerber, The representation theory of the symmetric group, Encyclopedia of Mathematics and its Applications, vol. 16