Abstract: Let H be a finite solvable group admitting an automorphism a of prime order p, 

here it is not assumed that p does not divide the order of H . We assume that a acts on H  in 

such  a  way that the order of any element of the semidirect product H  < a > lying outside H divides 

pq^m for some prime q # p and a fixed natural number m. A bound for the Fitting length of H  is 

determined under this condition. 

Key words: Finite Group, Solvable Group, Automorphism, Fitting length.