Generalized Snake lemma

Nope. Given any short exact sequence $0\to A\to B\to C\to 0$ mapping to $0\to A'\to B'\to C'\to 0$ you can extend to $0\to 0\to A\to B\to C\to 0$ mapping to $0\to D\to A'\oplus D\to B'\to C'\to 0$. Given $c\in C$ in the kernel of $C\to C'$, as usual we can map it to something in $A'$, but it won't be in the image of $D$ unless it's $0$.