Unipotent automorphisms of solvable groups

Let G be a solvable group and H a solvable subgroup of Aut ⁡ ( G ) $\operatorname{Aut}(G)$ whose elements are n-unipotent. When H is finitely generated, we show that it stabilizes a finite series in G and conclude that H is nilpotent. If G furthermore has a characteristic series with torsion-free factors. the same conclusion as above holds without the extra assumption that H is finitely generated.