On the influence of the fixed points of an automorphism to the structure of a group

Let α be a coprime automorphism of group G prime order and let P an α-invariant Sylow p-subgroup G. Assume that p∉π(CG(α)). Firstly, we prove is p-nilpotent if only CNG(P)(α) centralizes P. In the case Sz(2r) PSL(2,2r)-free where r=|α|, show p-closed CG(α) normalizes As consequence these two results, obtain G≅P×H for H We also generalization Frobenius p-nilpotency theorem groups admitting automorphisms order.