I have two groups $(Z_2)^n$ equiped with a pointwise binary addition operator and $S_n$ acting iteratively on the space $X=\{0,1\}^n$, in the following way: Fix $x\in X$. At each iteration we apply a transposition from $S_n$ to $x$ and then an element of $Z_2^n$ acts on the result, and so on. For example, fix $(1,0,0)\in X$, then for $(13)\in S_3$ and $(1,1,0)\in Z_2^3$, we have
$$(1,1,0) \circ ((13) \circ (1,0,0)) = (1, 1,0) \circ (0, 0,1) = (1,1,1)$$
Now choose other elements from $S_3$ and $Z_2^3$ And apply them to $(1,1,1)$. Continue this process.
How can one study this iterative process? Is there a way of mixing these two groups and define a group action?