Assume given is a smooth manifold $M$ and smooth vector bundles $A_1\subseteq A_2 \subseteq A_3$ such that $A_i$ is a subbundle of $A_{i+1}$ (embedded submanifold and a vector bundle). Is it generally true that $$\frac{A_3}{A_1} \cong \frac{A_3}{A_2} \bigoplus \frac{A_2}{A_1}?$$
It is easy to construct a homomorphism $\frac{A_3}{A_1} \to \frac{A_3}{A_2}$ and $\frac{A_2}{A_1} \to \frac{A_3}{A_1}$, but not $\frac{A_3}{A_2} \to \frac{A_3}{A_1}$ or $\frac{A_3}{A_1} \to \frac{A_2}{A_1}$. If one of the latter could be easily constructed, then using dimension arguments and such we would win. Is this possible in general?
I am not aware of any of the canonical maps above, unless we are in some contact/symplectic setting.