One point stabilizers in almost simple sharp permutation groups

In this work we consider primitive sharp permutation groups of type ({0,l}, n). According to a previous result of the author, either the structure of such a group (as well as the associated action) is completely determined, or the group is almost simple. We investigate the viability of the almost simple case, develope additional restrictions on the structure of one point stabilizers, and give some indication of a strategy for a proof that there are no almost simple, primitive sharp permutation groups of type ({0l}n).