Schur functions are irreducible characters of the unitary group $\mathcal{U}(N)$. This implies the integration formulae $$ \int_{\mathcal{U}(N)}s_\lambda(AUA^\dagger U^\dagger)dU=\frac{|s_\lambda(A)|^2}{s_\lambda(1)}$$ and $$ \int_{\mathcal{U}(N)}s_\lambda(AU)\overline{s_\mu(A U)}dU=\frac{\delta_{\lambda\mu}s_\lambda(AA^{\dagger})}{s_\lambda(1)},$$ where the overline means complex conjugation.

My question is whether we can compute the integral $$ \int_{\mathcal{U}(N)}s_\lambda(AUA^\dagger U^\dagger)\overline{s_\mu(AUA^\dagger U^\dagger)}dU$$