If a finite solvable group G admits an automorphism x of order p”, p some prime, such that x has no nontrivial fixed points on G, then the Fitting length of G is at most n. This result goes back to Hoffman [8], Shult [lo] and Gross [4,5], although in [8] and [lo] it is only proved under some additional hypothesis to the prime p and the primes dividing the order of G, and Gross could only show that the Fitting length is at most 2n - 2 for p = 2. The complete proof is contained, however, in [2], and Berger treats a much more general situation: the automorphism groups he allows do not have to be cyclic of prime power order, but only nilpotent. But let us still consider the automorphism x of order p”. We see, that the fixed point free action of x on G forces the structure of G to be “not too wild” in terms of the order of x. Now it is clear, that if x is allowed to have m fixed points on G, and (IGI, p) = 1, then also the Fitting length of G is bounded by some function of n and m, since the sections F,, ,,( G)/Fi- ,(G) must contain fixed points of x by the above theorem. But it seems more reasonable to look, whether G might still be “close to” having Fitting length at most n, differing from that only by some function of m. This was done in [6] and [7] for x of order p, i.e., it was shown that there is some function f such that (G: F(G)( is bounded by f (p, m), if x has order p, coprime to ICI, and lC,(x)l <m. We extend this result to automorphisms of prime power order and prove the following two theorems.