The permutation group $S_{2n}$ has $H_{2,n}=S_2\wr S_n$ as a subgroup. The plethysm $h_n(h_2)=\sum_{\lambda\vdash n}s_{2\lambda}$ is well known.

The zonal spherical functions $\omega_\lambda(g)=\frac{1}{H_{2,n}}\sum_{h\in H_{2,n}}\chi_{2\lambda}(gh)$ are responsible for the transition between power sum symmetric polynomials and zonal polynomials, $$Z_\mu \overset{\omega_\lambda(\mu)} \longrightarrow p_\lambda.$$

My question is what happens when we change the subgroup to $H_{n,2}=S_n\wr S_2$. The plethysm $h_2(h_n)=\sum_{(m_1,m_2)\vdash n}s_{(2m_1,2m_2)}$ is also well known. Are the corresponding zonal spherical functions known? Are they responsible for the transition between already studied clases of symmetric polynomials?