Let $B_n$ be the Boolean lattice and $G$ a subgroup of $S_n$ acting on $B_n$. Let $P_G=B_n/G$ denote the quotient poset.
Question 1: When is $P_G$ a lattice? When is it distributive in that case?
An example where it is not a lattice is for $n=5$ and the subgroup $G=langle(1,2,3,4,5)rangle$. But in this case it is still dissective, where dissective means that the MacNeille completion of the poset is a distributive lattice, see https://nreadin.math.ncsu.edu/papers/dissective.pdf .
Question 2: When is the poset $P_G$ dissective? When is it dissective but not a lattice?
I am not sure whether one can expect a general answer, but restrictions on G (cyclic, abelian…) or specific example of $P_G$ being dissective and not a lattice are also welcome