Any isomorphism implies a split exact sequence?

Question

For an exact sequence of modules $$ 0 \rightarrow A \xrightarrow{f} B \xrightarrow{g} C \rightarrow 0 $$ there is a map $\pi: B \rightarrow A$ for which $\pi \circ f = 1_A$ if and only if $A \oplus C$ is isomorphic to $B$ in such a way that $f$ corresponds to the canonical injection and $g$ corresponds to the canonical projection.

If a sequence is not split exact, is it still possible to have $A \oplus C \cong B$?

Answer 1

Yes, take the non-split exact sequence $$0 \rightarrow \mathbb{Z}\rightarrow\mathbb{Z} \rightarrow \mathbb{Z}_2 \rightarrow 0$$ Then look at the non-split exact sequence $$0 \rightarrow \mathbb{Z}\oplus\bigoplus_{\mathbb N}\mathbb{Z}_2\rightarrow \mathbb{Z}\oplus\bigoplus_{\mathbb N}\mathbb{Z}_2\rightarrow \mathbb{Z}_2 \rightarrow 0$$