So, I believe the following result is correct but do not know the exact reference (and not sure to what extent what I'm saying is true). If anyone could give a reference for this it would be great.

1) Consider three objects in some abelian category, $F_1, F_2, F_3$. Then objects with filtrations $F^1 \subset F^2 \subset F^3$ such that $F^1 = F_1, F^2/F^1 = F_2, F^3/F^2 = F_3$ are classified by triples $(\alpha, \beta, \gamma)$ of extension elements, $\alpha \in Ext^1(F_2, F_1), \beta \in Ext^1(F_3, F_2), \gamma \in Ext^1(F_3, F_1)$, such that the Yoneda product $\alpha \beta \in Ext^2(F_3, F_1)$ vanishes.

2) Similar result should hold for any amount of objects at least for the case of linear category in characteristic zero - the ways of gluing objects $F_1, ..., F_n$ into a filtered object $F^1 \subset F^2 \subset ... \subset F^n$ such that $F^k / F^{k-1} = F_k$ should be classified by Maurer-Cartan elements in the algebra $\bigoplus_{j>i}RHom^{\bullet}(F_j, F_i)$ (considered as either dgla or L-infinity algebra).

I also know that this type of questions frequently appear in the theory of mixed hodge structures (but unable to find any direct reference, too).

Edit: added forgotten $\gamma$