I have a clearly finite set of objects $S$ (single-string grammars of a strings $s$ over alphabet $\Sigma$, and a set of generators that generate what I call the smallest grammar algorithm group, $G$.
For the purpose of determining the size of $G$ (which might be huge), I would like to perform actions of the generators $g$ on the input string $s$. Yes, $Gs$ the orbit of $G$ on $s \in S$, is indeed all of $S$, so perhaps $S$ is not all single-string grammars, but only at least the ones we need to consider).
So for the purpose of computing $|G|$ and potentially $S$ itself, how do I enumerate the elements of $G$ by acting on $s$ with various combinations of the generators?
Is there a simple algorithm to do that?
The algorithm mentioned by @Qudit is called Schreier-Sims.
GAP and Magma have the algorithm built in, please try and let us know.