Let $G$ be a finite group with $|G|=n$, let $S_G=S_n$ be the group of $n!$ permutations of the set $G$. Then $G$ is a subgroup of $S_G$ via left-translation (i.e. $g\in G$ corresponds to the permutation $h \mapsto gh$).

My question: What can be said about the decomposition into irreducibles of the $S_G$-representation $\mathrm{Ind}_G^{S_G} 1$, where $1$ is the trivial $G$-representation? Is there some combinatorial way to identify the partitions $\lambda$ of $n$ corresponding to the irreducible $S_G$ representations $V_\lambda$ that appear in this induced representation?

Using Frobenius reciprocity, the representation $V_\lambda$ occurs iff restricted to $G$ it contains a copy of the trivial representation. The question can also be expressed by asking for which $\lambda$ we have $$\frac{1}{n} \sum_{g \in G} \chi_\lambda(h \mapsto gh)\neq 0,$$ where $\chi_\lambda$ is the character of $V_\lambda$. As the cycle-type of $h \mapsto gh$ is uniquely determined by $\mathrm{ord}(g)$, this shows that the answer of the question above only depends on the multiset $\{\mathrm{ord}(g): g \in G\}$.