This question follows up a previous question, Intersecting group orbits.

Suppose a group $G$ acts transitively on a set $X$ of $n$ elements, where $n$ is even, and consider the induced action on the set $\binom{X}{n/2}$ of subsets of $X$ of size $n/2$.

Question 1: Can there be precisely two orbits? More generally, if there is more than one orbit, how many orbits must there be?

Question 2: Suppose that no set is in the orbit of its complement. How many orbits must there be? Can there be precisely two orbits?

This question follows up a previous question, Intersecting group orbits.

Suppose a group $G$ acts transitively on a set $X$ of $n$ elements, where $n$ is even, and consider the induced action on the set $\binom{X}{n/2}$ of subsets of $X$ of size $n/2$.

Question 1: Can there be precisely two orbits? More generally, if there is more than one orbit, how many orbits must there be?

Question 2: Suppose that no set is in the orbit of its complement. How many orbits must there be? Can there be precisely two orbits?