Integration questions in separably good characteristics

Let G be a reductive group over an algebraically closed field k of separably good characteristic p>0 for G. Under these assumptions a Springer isomorphism from the reduced nilpotent scheme of the Lie algebra of G to the reduced unipotent scheme of G always exists. This allows to integrate any p-nilpotent element of Lie(G) into a unipotent element of G. One should wonder whether such a punctual integration can lead to a systematic integration of p-nil subalgebras of Lie(G). We provide counterexamples of the existence of such an integration in general as well as criteria to integrate some p-nil subalgebras of Lie(G) (that are maximal in a certain sense). This requires to generalise the notion of infinitesimal saturation first introduced by P. Deligne and to extend one of his theorem on infinitesimally saturated subgroups of G to the previously mentioned framework.