A Torsion-free Supersoluble Group with Trivial Outer Automorphism Group

We give a negative solution to Problem~13.23 of the Kourovka Notebook. We construct a torsion-free group $G$ of Hirsch length $14$ admitting a finite series $$1=G_0◃G_1◃\cdots ◃G_{14}=G$$ in which every $G_i$ is normal in $G$ and every factor is infinite cyclic, but such that $\text{Out}(G)=1$.