  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.
