Admissibility of Finite Groups over Semi-Global Fields

In 2009, Harbater, Hartmann, and Krashen gave a necessary and sufficient condition for a finite group G to be admissible over a semi-global field F so long as the characteristic of the corresponding residue field does not divide the order of G. They used a method known as field patching in order to show the sufficiency of their condition. Here we explore what happens when the characteristic of F (and also that of the residue field) divides the order of the group G. In particular, we show that if G is any p-group, where p is the characteristic of F, then it is admissible over F.