What are examples of expander family of 3-transitive Schreier graphs? Meaning for an action that is 3-transitive.

It is better to have an option for randomization. We know that choosing 2 elements at random in a simple Lie group leads to expander family of Cayley graphs.

Is the same thing true for example, in case we randomize elements in Schreier graph of $\mathrm{PSL}_2$ acting on the projective plane?

Let's ask this formally: Given $\epsilon$, is there a family of bounded degree Schreier graphs with a 3-transitive group action such that if I randomize x generators, I have $\epsilon$-expansion in probability p independent of n(the number of vertices of the graph).

I am looking for a known result similar to this. Or maybe references.

And if not 3, then 2-transitive would be OK.

Thanks