Characteristic classes of bundles of K3 manifolds and the Nielsen realization problem

Let K be the K3 manifold. In this note, we discuss two methods to prove that certain generalized Miller-Morita-Mumford classes for smooth bundles with fiber K are nonzero. As a consequence, we fill a gap in a paper of the first author, and prove that the homomorphism Diff(K) -> pi(0)Diff(K) does not split. One of the two methods of proof uses a result of Franke on the stable cohomology of arithmetic groups that strengthens work of Borel, and may be of independent interest.