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Let G be a finite group and A a group of automorphisms of G. Clearly A acts as a permutation group on G#, the set of non-identity elements of G. We assume that this permutation representation is half transitive, that is all the orbits have the same size. A special case of this occurs when A acts fixed point free on G. In this paper we study the remaining or non-fixed point free cases. We show first that G must be an elementary abelian g-group for some prime q and that A acts irreducibly on G. Then we classify all such occurrences in which A is a p-group.