We compute the auto-correlations functions of order $m\ge 1$ for the characteristic polynomials of random matrices from certain subgroups of the unitary groups $\U(2)$ and $\U(3)$ by applying branching rules. These subgroups can be understood as analogs of Sato--Tate groups of $\USp(4)$ in our previous paper. This computation yields symmetric polynomial identities with $m$-variables involving irreducible characters of $\U(m)$ for all $m \ge 1$ in an explicit, uniform way.