Quasi-endomorphism rings of some quasi-decomposable torsion-free Abelian groups of rank $4$

We obtain a description of quasi-endomorphism rings of torsion-free Abelian groups $G$ of rank $4$, quasi-decomposable into a direct sum of groups $A_1$ and $A_2$ of rank $1$ and a strongly indecomposable group $B$ of rank $2$ in the case where the quasi-homomorphism group $\mathbb {Q} \otimes \operatorname{Hom}(A_2,B)$ has rank $2$.