Let $F$ be a number field with ring of integers $o_F$. For a fixed prime number $p$ and $i\ge 2$ the étale wild kernels $W{K}_{2i-2}^{\acute{\mathrm{e}}\mathrm{t}}\left(F\right)$ are defined as kernels of certain localization maps on the $i$-fold twist of the $p$-adic étale cohomology groups of $\mathrm{spec}{o}_F[\frac{1}{p}]$. These groups are finite and coincide for $i=2$ with the $p$-part of the classical wild kernel $W{K}_2\left(F\right)$. They play a role similar to the $p$-part of the $p$-class group of $F$. For class groups, Galois co-descent in a cyclic extension $L/F$ is described by the ambiguous class formula given by genus theory....

Very little is known regarding the Galois group of the maximal $p$-extension unramified outside a finite set of primes $S$ of a number field in the case that the primes above $p$ are not in $S$. We describe methods to compute this group when it is finite and conjectural properties of it when it is infinite.