Modules over integer group rings of locally soluble groups with minimax restriction

Let $\mathbb Z$ be the ring of integers, $A$ be a $\mathbb ZG$-module, where $A/C_A(G)$ is not a minimax $\mathbb Z$-module, $C_G(A)=1$, and $G$ is a locally soluble group. Let $L_\mathrm{nm}(G)$ be the system of all subgroups $H\leq G$ such that quotient modules $A/C_A(H)$ are not minimax $\mathbb Z$-modules. The author studies $\mathbb ZG$-modules $A$ such that $L_\mathrm{nm}(G)$ satisfies the minimal condition as an ordered set. It is proved that a locally soluble group $G$ with these conditions is soluble. The structure of the group $G$ is described.