  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 $S^{2}\times S^{2}$. Furthermore, our
theorem also applies to certain exotic loops of diffeomorphisms on $S^{4}$ (as
recently discovered by Watanabe) and show that these loops have trivial family
Bauer--Furuta invariants. 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 $S^{4}$.
