Let $G$ be a finite permutation group acting transitively, but not regularly, on a set $V$. Let $H$ be the stabilizer of some point $v\in V$, and suppose that $H$ acts 2-transitively on one of its (non-trivial) orbits, i.e. $H$ acts 2-transitively on the (left) cosets of $H\cap H^g$ for an appropriate choice of $g\in G$.

General question: What can I conclude about the permutation character of $G$ acting on $V$?

Specifically:

- Can I give a lower bound for the multiplicities of the non-trivial irreducible components of the permutation character of $G$ on $V$?
- What about the dimensions of these components?
- What about if I weaken '2-transitively' to 'primitively' or something similar?