  In this paper we construct six-dimensional compact non-K\"ahler Hamiltonian
circle manifolds which satisfy the strong Lefschetz property themselves but
nevertheless have a non-Lefschetz symplectic quotient. This provides the first
known counter examples to the question whether the strong Lefschetz property
descends to the symplectic quotient. We also give examples of Hamiltonian
strong Lefschetz circle manifolds which have a non-Lefschetz fixed point
submanifold. In addition, we establish a sufficient and necessary condition for
a finitely presentable group to be the fundamental group of a strong Lefschetz
manifold. We then use it to show the existence of Lefschetz four-manifolds with
non-Lefschetz finite covering spaces.
