Infinite nonabelian corks

We construct $G$-corks for any extension $G$ of $\mathbb Z^m$ by any finite subgroup of $\mathrm{SO}(4)$ and weakly equivariant $G$-corks for any extension $G$ of $\mathbb Z^m$ by any finite solvable group. In particular, this is the first example of $G$-corks for an infinite nonabelian group $G$ and answers a question by Tange. The construction is a combination of previous results by Auckly-Kim-Melvin-Ruberman, Gompf, and Tange. Using Gompf's results about exotic $\mathbb R^4$'s, we give an application to construct exotic $\mathbb R^4$'s whose diffeotopy group contains all poly-cyclic groups.