On a property of split short exact sequences

Question

Let $A_{\bullet}, B_\bullet$ and $C_\bullet$ be three short exact sequences of groups (not necessarily abelian) out of which $A_\bullet$ and $B_\bullet$ are split. Assume that there is again a short exact sequence, $$0 \to A_{\bullet} \to B_{\bullet} \to C_{\bullet} \to 0$$of the short exact sequences. Is it necessary that $C_\bullet$ is also split?

Answer 1

$\newcommand{\ZZ}{\mathbb{Z}}$ Here is what I believe to be a counterexample, where all groups are $\ZZ$-modules (i.e., abelian groups):

$A_1 = 2 \ZZ$. $A_2 = \left\{\left(i,j\right) \in \ZZ \times \ZZ \mid 4 \mid 2i+j\right\}$ (the first "$\mid$" here is a "such that" symbol, while the second one is a "divides" sign). $A_3 = 2 \ZZ$. The injection $A_1 \to A_2$ sends each $i$ to $\left(i, 0\right)$, and the surjection $A_2 \to A_3$ sends each $\left(i, j\right)$ to $j$.

$B_1 = \ZZ$. $B_2 = \ZZ \times \ZZ$. $B_3 = \ZZ$. The injection $B_1 \to B_2$ sends each $i$ to $\left(i, 0\right)$, and the surjection $B_2 \to B_3$ sends each $\left(i, j\right)$ to $j$.

$C_1 = \ZZ / 2 \ZZ$. $C_2 = \ZZ / 4 \ZZ$. $C_3 = \ZZ / 2 \ZZ$. The injection $C_1 \to C_2$ sends each $\overline{i}$ to $\overline{2i}$, and the surjection $C_2 \to C_3$ sends each $\overline{i}$ to $\overline{i}$.

The maps $A_1 \to B_1$ and $B_1 \to C_1$ are the canonical inclusion and projection that one would expect. Same for the maps $A_3 \to B_3$ and $B_3 \to C_3$. The map $A_2 \to B_2$ is the canonical inclusion. The map $B_2 \to B_3$ sends each $\left(i, j\right)$ to $\overline{2i+j}$.

The sequences $A_\bullet$ and $B_\bullet$ are exact, and therefore are split because any short exact sequence which ends with a free module splits. But the sequence $C_\bullet$ is a non-split exact sequence.

That is, unless I've made a mistake, for which there is plenty of occasion...