Equivalence classes of (2,3)-pairs in PSL(2,q)

Question

Let $G = PSL(2,q)$. I am interested in the different ways of writing $G$ as an image of the modular group. If $A, B \in G$ are elements of order $2, 3$ respectively, then $\langle A,B \rangle$ is an image of the modular group (and is quite often the whole group $G$). Call $(A,B)$ a $(2,3)$-pair, and say that the pairs $(A,B)$, $(A',B')$ are equivalent if there is an automorphism of $G$ taking one pair to the other.

For example, I looked at the case $q = 5$ and there was exactly one equivalence class of $(2,3)$-pairs for each possible order $2, 3, 5$ of the element $AB$ (unless I made a mistake).

Is this true in general? If not, is there a nice way to list the equivalence classes in $G$?

Answer 1

[Thank you to Stefan Kohl for answering this at https://mathoverflow.net/q/182267/. I am reproducing it here as a Community Wiki answer.]

While your observation is true also for ${\rm PSL}(2,7)$, ${\rm PSL}(2,8)$ and ${\rm PSL}(2,9)$, it is false for ${\rm PSL}(2,11)$. -- In ${\rm PSL}(2,11)$ there are two orbits of $(2,3)$-pairs whose product has order $5$, as you can check with GAP as follows:

gap> G := PSL(2,11);;
gap> A := AutomorphismGroup(G);;
gap> pairs := Cartesian(Filtered(AsList(G),g->Order(g)=2),
>                       Filtered(AsList(G),g->Order(g)=3));;
gap> orbits := Orbits(A,pairs,OnTuples);;
gap> List(orbits,orbit->Order(Product(orbit[1])));
[ 5, 5, 11, 3, 2, 6 ]