Equivariant corks and Heegaard Floer homology

A cork is a contractible smooth 4-manifold with an involution on its boundary that does not extend to a diffeomorphism of the entire manifold. Corks can be used to produce exotic structures; in fact any two smooth structures on a closed simply-connected 4-manifold are related by a cork twist. Recently, Auckly-Kim-Melvin-Ruberman showed that for any finite subgroup of $\operatorname{SO}(4)$ there exists a contractible 4-manifold with an effective $G$-action on its boundary so that the twists associated to the elements of $G$ don't extend to diffeomorphisms of the entire manifold. We use a Heegaard Floer theory argument originating in work of Akbulut-Karakurt to reprove this fact.