Non-kinetic homotopy coherent actions on four-manifolds

We give the first example of a non-kinetic smooth homotopy coherent action of order two on a closed simply connected smooth four-manifold. This action is obtained by restricting a nontrivial smooth homotopy coherent action of the discrete circle group on a stabilized $K3$ surface. We also construct relatively non-kinetic smooth homotopy coherent extensions of boundary involutions over compact smooth $4$-manifolds. In addition, we exhibit boundary involutions that admit locally linear topological extensions but no smooth extensions over the same stabilized fillings. Nevertheless, we prove a Wall-type theorem showing that every free involution on a disjoint union of integral homology spheres extends smoothly over any simply connected smooth filling after sufficiently many stabilizations by $S^2\times S^2$.