Geometric automorphism groups of symplectic 4-manifolds

Let M be a closed, oriented, smooth 4-manifold with intersection form Γ, A(Γ) the automorphism group of Γ and D(M) the subgroup induced by orientation-preserving diffeomorphisms of M. In this note we study the question when D(M) is of infinite index in A(Γ) for a symplectic 4-manifold.