  Let S be a K3 surface. In part I of this paper, we constructed a
representation of the group Aut D(S), of auto-equivalences of the derived
category of S. We interpreted this infinite dimensional representation, as the
natural action of Aut D(S) on the cohomology of all moduli spaces of stable
sheaves (with primitive Mukai vectors) on S. The main result, of part I, is the
precise relation of this action with the monodromy of the Hilbert schemes S^[n]
of points on the surface. The proof of the above result was reduced, in part I,
to the case of two monodromy operators of S^[n], associated with choices of
line bundles on the surface S, of degree 2n-4 and 2n respectively. When n=1,
the first sequence of monodromy operators specializes to the reflection by a -2
curve. The n=1 case of the second sequence is related to the Galois involution,
of a double cover of the projective plane, branched along a sextic. We complete
the proof by treating these two sequences of examples.
