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This paper is the sequel to the paper “Commutative algebras of rational function matrices as endomorphisms of Kronecker modules I”. Here we focus on those Kronecker modules that are rank-2 extensions of finite-dimensional, torsion-free modules by the unique generic module R. We show that if such an extension is indecomposable, then its endomorphism algebra is trivial. This resolves the missing case from the previous paper, where the effects of height functions on the endomorphism algebras of extensions are studied.