On p-groups of maximal class as groups of automorphisms

Let P be a p-group of maximal class and M be a maximal subgroup of P. Let α   be an element in P∖MP∖M such that |CP(α)|=p2|CP(α)|=p2, and assume that |α|=p|α|=p. Suppose that P acts on a finite group G   in such a manner that CG(M)=1CG(M)=1. We show that if CG(α)CG(α) is nilpotent, then the Fitting height of G   is at most two and CG(α)CG(α) is contained in the Fitting subgroup of G  . For p=2p=2, without assuming that CG(α)CG(α) is nilpotent, we prove that the Fitting height h(G)h(G) of G   is at most h(CG(α))+1h(CG(α))+1 and the Fitting series of CG(α)CG(α) coincides with the intersection of CG(α)CG(α) with the Fitting series of G  . It is also proved that if CG(x)CG(x) is of exponent dividing e   for all elements x∈P∖Mx∈P∖M, then the exponent of G is bounded solely in terms of e   and |P||P|. These results are in parallel with known results on action of Frobenius and dihedral groups.