Fitting heights of odd-order groups with few character degrees

If G is any finite solvable group having a normal Sylow 2-subgroup (in particular, if G is odd) and satisfying cd(G) 5, where cd(G) is the set of ordinary irreducible character degrees of G, we show that the Fitting height of G does not exceed cd(G)−2. In case cd(G)=5, this upper bound on the Fitting height of G is best-possible.