Exotic diffeomorphisms on a contractible 4-manifold surviving two stabilizations

We develop a $\mathrm{P}\mathrm{i}\mathrm{n}(2)\times {\mathbb{Z}}_2$-equivariant refinement of the lattice homotopy type for computing equivariant Seiberg–Witten Floer homotopy types. As an application, we construct a relatively exotic diffeomorphism on a compact contractible 4-manifold that survives two stabilizations.