  This article is concerned with Galois theory for iterative differential
fields (ID-fields) in positive characteristic. More precisely, we consider
purely inseparable Picard-Vessiot extensions, because these are the ones having
an infinitesimal group scheme as iterative differential Galois group. In this
article we prove a necessary and sufficient condition to decide whether an
infinitesimal group scheme occurs as Galois group scheme of a Picard-Vessiot
extension over a given ID-field or not. In particular, this solves the inverse
ID-Galois problem for infinitesimal group schemes.
