  For suitable finite groups G, we construct contractible 4-manifolds C with an
effective G-action on $\partial C$ whose associated pairs (C,g) for all $g \in
G$ are distinct smoothings of the pair $(C,\partial C)$. Indeed C embeds in a
4-manifold so that cutting out C and regluing using distinct elements of G
yield distinct smooth 4-manifolds.
