Taking a group $G$ and a normal subgroup $G' \subseteq G$, we have a natural short exact sequence relating the two of them, namely: $$ 0 \longrightarrow G' \longrightarrow G \longrightarrow G/G' \longrightarrow 0.$$
We can see this sequence as a sequence attached to the (trivial) filtration induced by $G'$, that is $0\subseteq G' \subseteq G$.
Given a generic filtration $0\subseteq G_1 \subseteq G_2 \subseteq \cdots \subseteq G_n = G$, do we have a natural way to form a long exact sequence from it?
Same goes if we replace group by $A$-module for a certain ring $A$. Is there a way to construct a long exact sequence from a module filtration ?
No, there is not; long exact sequences and filtrations generalize short exact sequences in two different "directions" and are not directly related in general. Instead, from a filtration we just have a bunch of short exact sequences $$0\to G_k \to G_{k+1}\to G_{k+1}/G_k\to 0.$$