Suppose $R \to S$ is a homomorphism of rings (unital, but not necessarily commutative) and $\mathcal{C}$ is an abelian subcategory of the category of left $R$-modules such that the functor $S \otimes_R -$ is faithfully exact when restricted to $\mathscr{C}$. (This means that a short exact sequence in $\mathcal{C}$ is exact if and only if its image under $S \otimes_R -$ is exact. See Ishikawa's Faithfully Exact Functors... for a few standard equivalent formulations.)
Does it then follow that an exact sequence
$$0 \to A \to B \to C \to 0$$
in $\mathcal{C}$ is split if and only if
$$0 \to F(A) \to F(B) \to F(C) \to 0$$
is split?
The "only if" dirction is obvious, so the question is about the "if" direction. If $R$ and $S$ are commutative and every $M \in \mathcal{C}$ is finitely presented, this is true and is proved in Hochster's lecture notes here. It seems like a very formal statement, but Hochster's lecture notes, it's proved by considering the base change map
$$S \otimes_R \mathrm{Hom}_R(-,-) \to \mathrm{Hom}_S(S \otimes_R -, S \otimes_R -),$$
which doesn't make sense for non-commutative $R$ and $S$.
For what I have in mind, $R$ and $S$ are not commutative. But they are relatively nice, so if it helps to assume the rings are left noetherian or left coherent or something along those lines, that's just fine.