  In this paper we study the wild part of the finite monodromy groups of
abelian varieties over number fields. We solve Grunwald problems for groups of
the form $\mathbf{Z}/p\mathbf{Z}\wr \mathfrak{S}_n$ over number fields to build
CM abelian varieties with maximal wild finite monodromy in the odd prime case.
For the even prime case we prove a new bound on the 2-part of the order of the
finite monodromy group for CM abelian varieties and build varieties that reach
it.
