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 non-zero. As a consequence, we fill a gap in a paper of the first author, and prove that the homomorphism $Diff(K)\to {\pi }_{0}Diff(K)$ 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.