Homotopy Coherent Nielsen Realization Problem for Dehn Twists on K3-Type 4-Manifolds

We study the homotopy coherent version of the Nielsen realization problem for smooth $4$-manifolds. Given a finite subgroup $G\subset {\pi }_0(\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}(M))$, this problem asks whether there is a map $H\text{\hspace{0.17em}⁣}:BG\to B\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}(M)$ such that the induced map on fundamental groups coincides with the inclusion of $G$. Using family Seiberg-Witten theory, we prove that for $K3$-type $4$-manifolds, the Dehn twists along $(-2)$-spheres are not homotopy coherently Nielsen realizable. In particular, this gives an alternative proof of the failure of the classical Nielsen realization problem in this setting.