Trying to understand exact sequences better, I was searching for a simple context in which they naturally arise. (The exact sequences that occur in algebraic topology occur in a very complicated context.)

Somewhere, possibly on this site, I found the following example: Start with a formally infinite normal series of groups, $$ \dots \subset G_{n-1} \subset G_n \subset G_{n+1} \subset \cdots\,.$$ A finite sequence can be brought to this form by making the inclusions equalities in all but a finite number of cases.

From this series one can construct an exact sequence, $$ \dots \xrightarrow{\partial_{n-2}} A_{n-1} \xrightarrow{\partial_{n-1}} A_{n} \xrightarrow{\partial_{n}} A_{n+1} \xrightarrow{\partial_{n+1}} \cdots,$$ with $A_i = G_{i+2} / G_i$ and $\partial_{i}$ the function $g G_i \mapsto g G_{i+1}$.

Now I can think that an exact sequence is "really" a normal series, which is great for the intuition. However, I cannot invert the construction. If an exact sequence is given, how can I find the $G_i$, and how unique is such a reconstruction? (A common normal subgroup of all $G_i$ certainly gets lost in the exact sequence, but is this everything that can happen?)