Countably generated extensions of $QTAG$-modules

Let the $QTAG$-module $M$ be the set-theoretic union of a countable collection of isotype submodules $S_k$ of countable length. For $0\leqslant k <\omega$ we prove that $M$ is totally projective if $S_k$ is totally projective. Certain related assertions in this direction are also presented.