Let $V$ be our finite set of variables, and $T$ our finite set of terminals. Put $\Sigma = T \cup V$. Let $F = \text{FreeGroup}(\Sigma)$. I want to represent all nice enough CFG's "over $F$" of any language over $T$ as a subgroup of $X = \bigoplus_{A \in V}\bigoplus_{i=1}^{\infty} F$.
For each $x \in X$, the $(A, i)$th component corresponds to the $i$th rule for the variable $A$, i.e. $A \to x_{A, i}$, other wise that component has the value $1$ e.g. when the grammar has no such rule.
We will call grammar nice enough if it "sees all of its intended variables starting at $S \in V$." An intended variable $A$ of $x$ is one for which $x_{A, i} \neq 1$ for some $i \in \Bbb{N}$. $x$ will see the variable $A$ if $A \in \text{Vars}_x(x(S))$, where:
$$ \begin{cases} \text{Vars}_x(1) = \varnothing, \\ \text{Vars}_x(a)= \varnothing, \ \ a \in T \cup T^{-1}\\ \text{Vars}_x(A) = \{A\} \cup \bigcup_{i\in I(x, A)}\text{Vars}_x(x_{A, i}), \ \ A \in V \cup V^{-1} \\ \text{Vars}_x(zy) = \text{Vars}_x(z) \cup \text{Vars}_x(y), \ \ z, y \in F\\ \end{cases} $$ , where $I(x, A) = \{ i \in \Bbb{N} : x_{A, i} \neq 1\}$.
Do nice enough grammars form subgroup of $X$?
The operation in $F$ is componentwise concatenation, by the way, not usual CFG concatentation of the starting rules.