The question is natural. Let's just work in the category of modules over a ring. Pick three modules $M_1, M_2, M_3$. Consider consecutive extensions of these modules, i.e., consider M, such that we have short exact sequences $0 \to M_1 \to M \to M' \to 0$, and $0 \to M_2 \to M' \to M_3 \to 0$. This is a naturally defined functor right? And do all these extensions form an abelian group? (as for the extension of two modules). And what are other properties for these extensions?
I suppose this should be some standard materials in homological algebra, but I could not find any references.
Thanks!