I would like some examples of groups $G$ satisfying all of the following criteria:

- $G<Sym(n)$, the symmetric group on $n$ letters, and $G$ is primitive.
- $G$ has a regular suborbit, i.e. if $M$ is a point-stabilizer, then there is an orbit of $M$ on which $M$ acts regularly.
- $G$ has a non-faithful suborbit, i.e. if $M$ is a point-stabilizer, then there is an orbit of $M$ on which $M$ acts non-faithfully.

Just to be clear: the suborbits in 2 and 3 need to be different! If possible, I would like to strengthen 3:

- $G$ has a non-faithful suborbit $\Lambda$ such that, if $M$ is the associated point-stabilizer, then $M^\Lambda\cong Alt(\Lambda)$ or $Sym(\Lambda)$. Here $M^\Lambda$ is the permutation group induced on $\Lambda$ by the action of $M$.

I don't know how common such groups are... So for now I'm just interested in collecting different examples to see if I can pick out some interesting properties. GAP/ MAGMA computations, as well as theoretical answers telling me what such groups might look like, are of interest to me.

Thanks in advance.