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