Tableaux problem 29

29.[Stan1999a]*Ch. 7 Exer. 71(e) Given a finite group $G$, define $\psi_{G}$ as the character of the action of $G$ on itself by conjugation. That is, the character of permutation representation $\rho:G\rightarrow\mathfrak{S}_{G}$ defined by $\rho(x)(y)=xyx^{-1}$, where $\mathfrak{S}_{G}$ is the group of permutations of $G$. Define $\kappa_{n\lambda}=\left<\psi_{\mathfrak{S}_{n}},\chi^{\lambda}\right>$, where $\chi^{\lambda}$ is the irreducible representation of $\mathfrak{S}_{n}$ indexed by the partition $\lambda$. $\kappa_{n\lambda}>0$ with the sole exception of $n=2,\lambda=(1,1)$. Is there a “nice” combinatorial interpretation of the numbers $\kappa_{n\lambda}$?