Induced Hamiltonian maps on the symplectic quotient

Let (M,ω)(M,ω) be a symplectic manifold and G a compact Lie group that acts on M. Assume that the action of G on M is Hamiltonian. Then a G-equivariant Hamiltonian map on M induces a map on the symplectic quotient of M by G. Consider an autonomous Hamiltonian H with compact support on M  , with no non-constant closed trajectory in time less than 1 and time-1 map fHfH. If the map fHfH descends to the symplectic quotient to a map Φ(fH)Φ(fH) and the symplectic manifold M   is exact and Ham(M,ω)Ham(M,ω) has no short loops, we prove that the Hofer norm of the induced map Φ(fH)Φ(fH) is bounded above by the Hofer norm of fHfH.