The paper of Hartley

A conjecture of Bachmuth and Mochizuki on automorphisms of soluble groups. Canad. J. Math. 28 (1976), no. 6, 1302-1310.

provides many counterexamples.

Let me quote from MathSciNet review:

J. Tits [J. Algebra 20 (1972), 250--270; MR0286898 (44 #4105)] showed that a finitely generated linear group either contains a soluble subgroup of finite index or else contains a nonabelian free subgroup. S. Bachmuth and H. Y. Mochizuki [Bull. Amer. Math. Soc. 81 (1975), 420--422; MR0364452 (51 #706)] conjectured that a finitely generated group of automorphisms of a finitely generated soluble group $G$ satisfies the same conclusion. The conjecture is known to be true when $G$ is nilpotent-by-abelian. This paper shows that the conjecture is false in general and a family of counterexamples is constructed. In particular there are such examples where $G$ has derived length three.

The paper of Hartley

A conjecture of Bachmuth and Mochizuki on automorphisms of soluble groups. Canad. J. Math. 28 (1976), no. 6, 1302-1310.

provides many counterexamples.

Let me quote from MathSciNet review:

J. Tits [J. Algebra 20 (1972), 250--270; MR0286898 (44 #4105)] showed that a finitely generated linear group either contains a soluble subgroup of finite index or else contains a nonabelian free subgroup. S. Bachmuth and H. Y. Mochizuki [Bull. Amer. Math. Soc. 81 (1975), 420--422; MR0364452 (51 #706)] conjectured that a finitely generated group of automorphisms of a finitely generated soluble group $G$ satisfies the same conclusion. The conjecture is known to be true when $G$ is nilpotent-by-abelian. This paper shows that the conjecture is false in general and a family of counterexamples is constructed. In particular there are such examples where $G$ has derived length three.

The paper of Hartley

A conjecture of Bachmuth and Mochizuki on automorphisms of soluble groups. Canad. J. Math. 28 (1976), no. 6, 1302--1310

provides many counterexamples.

Let me quote from MathSciNet review:

"J. Tits [J. Algebra 20 (1972), 250--270; MR0286898 (44 #4105)] showed that a finitely generated linear group either contains a soluble subgroup of finite index or else contains a nonabelian free subgroup. S. Bachmuth and H. Y. Mochizuki [Bull. Amer. Math. Soc. 81 (1975), 420--422; MR0364452 (51 #706)] conjectured that a finitely generated group of automorphisms of a finitely generated soluble group $G$ satisfies the same conclusion. The conjecture is known to be true when $G$ is nilpotent-by-abelian. This paper shows that the conjecture is false in general and a family of counterexamples is constructed. In particular there are such examples where $G$ has derived length three".

The paper of Hartley

A conjecture of Bachmuth and Mochizuki on automorphisms of soluble groups. Canad. J. Math. 28 (1976), no. 6, 1302-1310.

provides many counterexamples.

Let me quote from MathSciNet review:

J. Tits [J. Algebra 20 (1972), 250--270; MR0286898 (44 #4105)] showed that a finitely generated linear group either contains a soluble subgroup of finite index or else contains a nonabelian free subgroup. S. Bachmuth and H. Y. Mochizuki [Bull. Amer. Math. Soc. 81 (1975), 420--422; MR0364452 (51 #706)] conjectured that a finitely generated group of automorphisms of a finitely generated soluble group $G$ satisfies the same conclusion. The conjecture is known to be true when $G$ is nilpotent-by-abelian. This paper shows that the conjecture is false in general and a family of counterexamples is constructed. In particular there are such examples where $G$ has derived length three.