In "Group Extensions and Homology" Eilenberg and Mac Lane define
$$Ext\ (G,H) =Fact\ (G,H)/Trans\ (G,H)$$
for a normal subgroup $H$ that extends the group $G$. The trivial case of $H$ that extends the group $G$ is for example: $$ 1 \rightarrow (\mathbb{R}, \cdot) \rightarrow (\mathbb{R} \times \mathbb{R}, \cdot) \rightarrow (\mathbb{R}, \cdot) \rightarrow 1$$
Now, E & M also define a transformation as satisfying $$ t(h,k)=g(h)+g(k)-g(h+k) $$ for a group under addition where $h,k \in H$ and $g(h), g(k) \in G$
Allowing $f(x)=x$ to be a function in $(\mathbb{R} \times \mathbb{R}, \cdot)$ above, this gives me $1$, making $Trans(G,H)$ the trivial group which makes sense if (as other sources say) the transformation group is the permutation group or group of automorphisms.
Now my problem is this correlates to the following for me, which makes no sense $$Ext\ (G,H) =Fact\ (G,H)/Trans\ (G,H)$$ $$(\mathbb{R}\times\mathbb{R}, \cdot) = (\mathbb{R}, \cdot)/(1,\cdot)$$
What am I doing wrong in terms of the groups? Also, I have no reason to believe it does, but if my mistake relates to homology, I should warn you that I do not understand homology yet.
As Prof. Arturo Magidin kindly pointed out, $Ext$, $Trans$, and $Fact$ represent the collections whose elements satisfy Mac Lane and Eilenberg's definitions for and extension of $G$ by $H$ where $h,k,l\in H$ and $$t(h,k)\in Fact$$ $$t(h,k)=g(h)+g(k)-g(h+k)$$ Which, these $t$ always satisfy the conditions for the elements $f \in Fact$ s.t. $$f(h,k) +f(h+k,l)=f(h,k+l)+f(k,l) \\ f(h,k) =f(k,h)$$
Where, these collections form groups. But, the groups formed are not to be confused with common notions of transformation groups and factor (or quotient) groups.
This gives the result for $\mathrm{Ext}\{G,H\}$ the group of group extensions of $G$ by $H$ for $\mathrm{Fact}\{G,H\}$ group of all factor sets and $\mathrm{Trans}\{G,H\}$ group of all transformation sets.
$$\mathrm{Ext}\{G,H\} = \mathrm{Fact}\{G,H\} / \mathrm{Trans}\{G,H\}$$
I answered this to close the question and give a succinct response for future readers, but all credit of explanation goes to Prof. Arturo Magidin.