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 $A \oplus C \cong B$?