Murphy's Law for Galois Deformation Rings

In this paper, we prove, under a technical assumption, that any semi-direct product of a $p$-group $G$ with a group $\mathrm{\Phi }$ of order prime to $p$ can appear as the Galois group of a tower of extensions $H\mathrm{/}K\mathrm{/}F$ with the property that $H$ is the maximal pro-$p$ extension of $K$ that is unramified everywhere, and $\mathrm{Gal}(H\mathrm{/}K)=G$. A consequence of this result is that any local ring admitting a surjection to ${\mathbb{Z}}_5$ or ${\mathbb{Z}}_7$ with finite kernel can occur as a universal everywhere unramified deformation ring.