Inverse Galois problem on the upper or lower numbering filtration

Question

Let $K$ be a (complete) discrete valuated field and $E$ a Galois extension of group $G$. Then one has two filtrations on $G$, the upper and the lower numbering.

Assume that $K$ is equal to its maximal tame extension (that is, forget about the $0$-th part of these filtrations) and that the residue field of $ K$ has characteristic $p>0$, then my questions are: if one fixes $G$ a $p$-group and an arbitrary filtration $G_i$ of normal subgroups (possibly with $G_i=G_{i+1}$ for some $i$) with $G_i/G_{i+1}$ abelian, is there $K$ and $E$ as before such that this is the lower numbering filtration of $E/K$? What about the upper?

Are there further restrictions that ensure such an existence?

Answer 1

It is not sufficient to require that the quotients $G_i/G_{i+1}$ be commutative; they also have to have exponent dividing $p$. There are further restrictions, and the question has been studied by Maus in particular.

See for example

Eckart Maus, Die gruppentheoretische Struktur der Verzweigungsgruppenreihen, Journal für die reine und angewandte Mathematik (1968) Volume: 230, page 1-28 https://eudml.org/doc/150845

Review : http://www.ams.org/mathscinet-getitem?mr=225763

Here is a summary in English :

Eckart Maus, On the jumps in the series of ramifications groups, Mémoires de la Société Mathématique de France (1971) Volume: 25, page 127-133 https://eudml.org/doc/94571

Review : http://www.ams.org/mathscinet-getitem?mr=364194