This is a problem that I thought at first it is obvious but it became less clear the more I thought about it. Assume we have a finitely generated algebra $A$ over a field $k$. A short exact sequence of projective $A$-modules $l_1:0\rightarrow P_1 \rightarrow P_2 \rightarrow P_3 \rightarrow 0$. Let's assume we can lift this short exact sequence to their generators, i.e. let $F_i$ be the free module that surjects onto $P_i$ and assume there is a short exact sequence $l_2: 0\rightarrow F_1 \rightarrow F_2 \rightarrow F_3 \rightarrow 0$ that surjects onto $l_1$. Furthermore assume that all morphisms in $l_2$ are given by matrices in $M_{n\times m}(k)$ for suitable values of $m$ and $n$. This implies that we can choose splitting morphisms for $l_2$ that are also in $M_{n\times m}(k)$. My question: is there a such a choice of a splitting for $l_2$ that induces a splitting on $l_1$? This is equivalent to asking whether such a splitting will map the kernels of $F_i\rightarrow P_i$ to each other. (If answer depends on $A$ for what $A$'s this can be true?)