Permutation groups where non-trivial elements have few fixed points

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During the first phase of our cooperation Kay Magaard and I produced a few good results and some more precise questions emerged that we plan to answer in 2014.The basis of our project is a result of Schoeneberg's. It says that all fixed points of a non-trivial automorphism of an algebraic curve that fixes at least five points is a Weierstrass point.With this theorem in mind, Kay Magaard and I started to investigate permutation groups where every non-trivial element has few fixed points, hence being an obstruction to finding Weierstrass points with Schoeneberg's result. It is our aim to understand the structure of such groups and to classify the simple groups with this property completely.