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.