Finite homomorphic images of groups of finite rank

Let $\pi$ be a finite set of primes. We prove that each soluble group of finite rank contains a finite index subgroup whose every finite homomorphic $\pi$-image is nilpotent. A similar assertion is proved for a finitely generated group of finite rank. These statements are obtained as a consequence of the following result of the article: Each soluble pro-$\pi$-group of finite rank has an open normal pronilpotent subgroup.