Let for any sets $A,B,C$ having the same algebraic structure (they can be groups, rings, modules, etc.) $$0 \to A \to B \to C \to 0$$ be a short exact sequence with $f: A \to B$ and $g: B\to C$ s.t $Im (f) = Ker (g)$.
It is clear that $B$ has a substructure that is algebraically equivalent (isomorphic) to $A$ s.t when we consider the quotient of that substructure with $B$, it is algebraically equivalent (isomorphic) to $C$.
Is there any algebraic structure and a operation between such structures that we can "construct" $B$ out of $A$ and $C$ ? In other words, is there any algebraic structure and an operation on the set of all such algebraic structures, denote $+$, $A+C$ is algebraically equivalent to $B$ ?
Edit:
Note that, I looking for something a general way of "constructing" $B$ out of $A$ and $C$ s.t we can do this "construction" what ever $A,B,C$ is as long as they have that algebraic structure.
There cannot be the operation that you wish on the nose, because, as Andreas Blass points out in the comments, in most of these categories there are non-trivial extensions, i.e. there is more than one way to fit an object $\square$ in an exact sequence $$ 0 \to A \to \square \to C \to 0. $$
However, you can make a group out of an "exact category" (the so called Grothendieck group, where "exact category" roughly means there's enough structure to make sense of short exact sequences in the category). The idea is to take as elements of the group all (isomorphism classes of) objects and formally identifying $A + C = B$ whenever $$ 0 \to A \to B \to C \to 0. $$ See (https://en.wikipedia.org/wiki/Grothendieck_group#Grothendieck_group_and_extensions) for details.