Let $G$ be a finite group. When does the automorphism group $\text{Aut}(G)$ act transitively on a subset of $n$-tuples of generators of $G$ that multiply to 1?
I'm interested in this question in relation to the triangle groups. For example, when $G=\text{PSL}(2,7)$, there are exactly 336 triples of generators of orders 2, 3, and 7 which multiply to 1. In this case $\text{Aut}(G)=\text{PGL}(2,7)$ and so $|\text{Aut}(G)|=|\text{PGL}(2,7)|=336$. In other cases $|\text{Aut}(G)|$ divides the number of triples. Is the action still transitive in these cases?