Summary: The concept of QTAG-modules was introduced by S. Singh [Acta Math. Hung. 50, 85-95 (1987; Zbl 0628.16014)]. A module \(M\) is said to be a QTAG-module if every finitely generated submodule of every homomorphic image of \(M\) is a direct sum of uniserial modules. Various mathematicians generalized many results of primary Abelian groups for these modules. Totally projective QTAG-modules have significance in the theory of modules, which may be generalized in several ways. Here we study them in terms of nice decomposition series. W. Liebert [Abelian group theory, Proc. Conf., Honolulu 1983, Lect. Notes Math. 1006, 384-399 (1983; Zbl 0531.20032)] defined a rich group as a \(p\)-group \(G\) if every height-increasing homomorphism from a finite subgroup of \(G\) into \(G\) can be extended to an endomorphism of \(G\). In this paper we generalize this concept for QTAG-modules and further characterise the endomorphism ring of the rich decent QTAG-modules.