  In this second part of the paper, we consider finite difference Lagrangians
which are invariant under linear and projective actions of $SL(2)$, and the
linear equi-affine action which preserves area in the plane.
  We first find the generating invariants, and then use the results of the
first part of the paper to write the Euler--Lagrange difference equations and
Noether's difference conservation laws for any invariant Lagrangian, in terms
of the invariants and a difference moving frame. We then give the details of
the final integration step, assuming the Euler Lagrange equations have been
solved for the invariants. This last step relies on understanding the Adjoint
action of the Lie group on its Lie algebra. We also use methods to integrate
Lie group invariant difference equations developed in Part I.
  Effectively, for all three actions, we show that solutions to the
Euler--Lagrange equations, in terms of the original dependent variables, share
a common structure for the whole set of Lagrangians invariant under each given
group action, once the invariants are known as functions on the lattice.
