On the quasi-endomorphism rings of quasi-decomposable torsion-free Abelian groups of rank $4$ with a strongly indecomposable quasi-summand of rank $2$

We obtain a description of quasi-endomorphism rings of torsion-free Abelian groups $G$ of rank $4$ that are 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 $\mathbb {Q} \otimes \operatorname{Hom}(B, A_2)$ has rank $2$.