Let $A$, $B$, and $C$ be Abelian groups (more generally, $R$-modules. Even more generally, objects of some Abelian category. I will suppress any decorations on my $\text{Ext}$ groups.) I'm interested in "extensions of (extensions of A by B) by C", that is, extensions of the form $$ 0 \to C \to Y \to X \to 0$$ where $X$ is an extension of $A$ by $B$: $$ 0 \to B \to X \to A \to 0.$$
Denote the set of these by $\text{Ext}(A,B,C)$. Here are some things I know about this set:
This is "associative" in $A$, $B$, and $C$. More precisely, ''extensions of (extensions of $A$ by $B$) by $C$" are naturally in bijection with "extensions of $A$ by (extensions of $B$ by $C$)". The correspondence is as follows: Given $X$ and $Y$ above, pull back $B$ along $Y \to X$, this is a sub of $Y$ with quotient $A$ and it is an extension of $B$ by $C$. Conversely, given a
$$0 \to X \to Y \to A \to 0$$ $$0 \to C \to X \to B \to 0$$
we can take the pushout of $B$ along $X \to Y$ to get an object of the first type.
My questions can be summarized with: what kind of structure does the set $\text{Ext}(A,B,C)$ have? My original hope would be that it has the structure of an Abelian group in such a way that the maps $\text{Ext}(A,B,C) \to \text{Ext}(A,B)$ and $\text{Ext}(A,B,C) \to \text{Ext}(B,C)$ (defined by $Y \mapsto X$ in the above notation) are homomorphisms, but this seems like it might be too good to be true.
One thing I can say about $\text{Ext}(A,B,C)$ is that it is somehow a collection of "groups fibered over a group" in two different ways: We have the map $\text{Ext}(A,B,C) \to \text{Ext}(A,B)$. The codomain is a group, and the fiber above an element $X \in \text{Ext}(A,B)$ is the group $\text{Ext}(X,C)$, and there is a similar statement for the projection $\text{Ext}(A,B,C) \to \text{Ext}(B,C)$. Furthermore, the intersection of the fibers above the identity elements in $\text{Ext}(A,B)$ and $\text{Ext}(B,C)$ is canonically $\text{Ext}(A,C)$, but I'm not sure what I can say about the intersection of more general fibers.
In terms of functoriality, I believe that $\text{Ext}(A,B,C)$ is contravariant in $A$ and covariant in $C$, but I can't seem to make it functorial in $B$, and the issue that arises seems to be the same issue that arises when I try and put a group structure on this set.
So, my questions:
1. Is $\text{Ext}(A,B,C)$ a group in a reasonable way?
2. Is it functorial in $A$, $B$, $C$?
3. Is it a homological object? That is, does it arise in a similar way to the regular $\text{Ext} functors?
Now that I've typed this out, I think there's a map $\text{Ext}(A,B,C) \to \text{Ext}^2(A,C)$ defined by sending the extension $$0 \to C \to Y \to X \to 0$$ $$0 \to B \to X \to A \to 0$$
to the $2$-extension $$0 \to C \to B' \to X \to A \to 0$$ where $B'$ is the pullback of $B$ along $Y \to X$.
Is there anything that can be said about the image of this map as it depends on $B$?