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We establish the existence of a pair of exotic surfaces in a punctured $K3$which remains exotic after one external stabilization and have diffeomorphiccomplements. A key ingredient in the proof is a vanishing theorem of the familyBauer--Furuta invariant for diffeomorphisms on a large family of spin4-manifolds, which is proved using the tom Dieck splitting theorem inequivariant stable homotopy theory. In particular, we prove that the$S^{1}$-equivariant family Bauer--Furuta invariant of anyorientation-preserving diffeomorphism on $S^{4}$ is trivial and that the$\mathrm{Pin}(2)$-equivariant family Bauer--Furuta invariant for adiffeomorphism on $S^{2}\times S^{2}$ is trivial if the diffeomorphism actstrivially on the homology. Therefore, these invariants do not detect exoticself-diffeomorphisms on $S^{4}$ or $S^{2}\times S^{2}$. Furthermore, ourtheorem also applies to certain exotic loops of diffeomorphisms on $S^{4}$ (asrecently discovered by Watanabe) and show that these loops have trivial familyBauer--Furuta invariants. En route, we observe a curious element in the$\mathrm{Pin}(2)$-equivariant stable homotopy group of spheres which couldpotentially be used to detect an exotic diffeomorphism on $S^{4}$.

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