Family Bauer--Furuta invariant, Exotic Surfaces and Smale's conjecture

We establish the existence of a pair of exotic surfaces in a punctured $K3$ which remains exotic after one external stabilization and have diffeomorphic complements. A key ingredient in the proof is a vanishing theorem of the family Bauer--Furuta invariant for diffeomorphisms on a large family of spin 4-manifolds, which is proved using the Tom Dieck splitting theorem in equivariant stable homotopy theory. In particular, we prove that the $S^{1}$-equivariant family Bauer--Furuta invariant of any orientation-preserving diffeomorphism on $S^{4}$ is trivial and that the $\mathrm{Pin}(2)$-equivariant family Bauer--Furuta invariant for a diffeomorphism on $S^{2}\times S^{2}$ is trivial if the diffeomorphism acts trivially on the homology. Therefore, these invariants do not detect exotic self-diffeomorphisms on $S^{4}$ or $S^{2}\times S^{2}$. En route, we observe a curious element in the $\mathrm{Pin}(2)$-equivariant stable homotopy group of spheres which could potentially be used to detect an exotic diffeomorphism on $S^{4}$.