Suppose that $S \subset R$ is a ring extension, and $T$ is a flat extension of $R$. Consider an exact sequence of $R$-modules: $$ 0\to X_1\to X_2\to X_3\to 0, $$ which splits as a sequence of $S$-modules. Under what conditions on the extension $S \subset R$ and the module $T$ one can ensure that the induced sequence $$ 0\to T\otimes_R X_1\to T\otimes_RX_2\to T\otimes_R X_3\to 0, $$ also splits as a sequence of $S$-modules?
Some conceivable conditions include: $S$ being semisimple; $S=R$; and $T$ being a finitely generated projective $R$-module. Are there any other noteworthy cases?