Five term exact sequence - basis for the middle when I understand the rest?

Question

Suppose I have an exact sequence of say finite dimensional $\mathbb{Z}/2$ vector spaces $$ A \to B \to C \to D \to E $$ and suppose I have bases for $A,B,D,E$ and I completely understand the maps $A \to B$ and $D \to E$, but I do not know what $C$ is (only that it fits into the above exact sequence). Can I write down in some way (probably involving some choices) a basis for $C$ where the elements in some way defined by either there preimage (if they have one) with the map $A \to B$, or there image with the map $D \to E$.

Answer 1

Denoting the mapping $A\rightarrow B$ by $\alpha$, $B\rightarrow C$ by $\beta$, $C\rightarrow D$ by $\gamma$ and $D\rightarrow E$ by $\delta$, there is the induced short exact sequence $$0\rightarrow B/\ker \beta\rightarrow C\rightarrow {\rm im}~\gamma\rightarrow 0.$$ Since $\ker \beta={\rm im ~}\alpha$ and $\ker \delta={\rm im}~\gamma,$ a basis of $C$ can be taken of the form (by abuse of notations) $$\beta(B/{\rm im}~\alpha)\cup \gamma^{-1}(\ker\delta),$$ where $B/{\rm im}~\alpha$ denotes a set of linearly independent vectors in $B$ which remain independent in the quotient, and $\gamma^{-1}(\ker \delta)$ is a preimage (which involves a choice) of a set of linearly independent vectors in $\ker\delta$.