Here is a paper by Rachel Pries and Katherine Stevenson that addresses the question:
http://www.math.colostate.edu/~pries/Preprints/11dgroupreportv036.pdf
Full reference: "A survey of Galois theory of curves in characteristic p" , Fields Institute Communications 60 , American Mathematical Society, Providence RI, (2011), 169-191.
I am not well-versed in this area, but I found this paper quite readable. Some of the examples in the paper have been brought up in answers already in this thread.