Let $A, B, C$ be holomorphic vector bundles over some complex manifold $X$. Let $A', B', C'$ be sub bundles, respectively. Suppose that we have short exact sequences:
$$0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$$
$$0 \rightarrow A' \rightarrow B' \rightarrow C' \rightarrow 0$$
Let's also suppose that the quotient sequence
$$0 \rightarrow A / A' \rightarrow B / B' \rightarrow C / C' \rightarrow 0$$
is also exact. For example, the map between sub bundles $A' \rightarrow B'$ is induced from the map $A \rightarrow B$.
If the first sequence is in fact split, can one conclude that the quotient sequence is also split? If not, what further conditions does one need to impose for it to be split?
No, in fact any short exact sequence $0\to X\stackrel{\alpha}{\to}Y\stackrel{\beta}{\to}Z\to0$, split or not, is isomorphic to the quotient of a split short exact sequence by a split short exact subsequence, as follows:
$$\require{AMScd}\begin{CD} @[email protected]@.0\\ @.@VVV@VVV@VVV\\ 0@>>>0@>>>X@>1>>X@>>>0\\ @.@VVV@VV\begin{pmatrix}1\\\alpha\end{pmatrix}V@VV\alpha V\\ 0@>>>X@>\begin{pmatrix}1\\0\end{pmatrix}>>X\oplus Y@>\begin{pmatrix}0&1\end{pmatrix}>>Y@>>>0\\ @.@VV1V@VV\begin{pmatrix}\alpha&-1\end{pmatrix}V@VV-\beta V\\ 0@>>>X@>\alpha>>Y@>\beta>>Z@>>>0\\ @.@VVV@VVV@VVV\\ @[email protected]@.0 \end{CD}$$