For any K¨ahler surface which admits no nonzero holomorphic vectorfields, we consider the group of holomorphic automorphisms which induce identity on the second rational cohomology. Assuming the canonical linear system is without base points and fixed components, C.A.M. Peters [12] showed that this group is trivial except when the K¨ahler surface is of general type and either c21 = 2c2 or c21 = 3c2 holds. Moreover, this group is a 2-group in the former case, and is a 3-group in the latter. The purpose of this note is to give further information about this group. In particular, we show that c21 is divisible by the order of the group. Our argument is based on the results of C.H. Taubes in [14, 15] on symplectic 4-manifolds, which are applied here in an equivariant setting.