  Given a Lie group $G$, we elaborate the dynamics on $T^*T^*G$ and $T^*TG$,
which is given by a Hamiltonian, as well as the dynamics on the Tulczyjew
symplectic space $TT^*G$, which may be defined by a Lagrangian or a Hamiltonian
function. As the trivializations we adapted respect the group structures of the
iterated bundles, we exploit all possible subgroup reductions (Poisson,
symplectic or both) of higher order dynamics.
