Let $X$ be a finite set and $G$ be a transitive subgroup of the symmetric group on $X.$ Recall that a (complete) block system for this action is a partition of $X = B_1 \cup \cdots \cup B_k$ into equally-sized blocks $B_1 , \ldots , B_k,$ which is preserved in the sense that blocks are always mapped to blocks under the group action. The block systems for a particular action form a lattice whose maximum is $\{ X \}$ and whose minimum is the partition of $X$ into singletons.
In the lattice of block systems, we may associate to any chain a sequence of integers: starting from the maximum element of that chain, divide the size of a block of the given block system by the size of a block of its predecessor.
Question: Is there a name for group actions or permutation groups satisfying the condition that the aforementioned sequences associated to every maximal chain determine the same multiset?
Example 1: Let $G=X=A_4$ acting on itself by multiplication. The lattice of block systems is essentially the lattice of subgroups (blocks $=$ cosets.) There are two incomparable chains of subgroups whose associated sequences are $(3, 2, 2)$ and $(4, 3),$ so the condition is not satisfied.
Example 2: $G = A_4$ acting as usual on $ X= \{ 1,2,3,4 \}$ is primitive, so the only sequence is $(4)$ and the condition is trivially satisfied.
Example 3: For $D_8$ acting on itself by multiplication, the sequence is always $(2, 2, 2),$ so the condition is satisfied.
Example 4: For $\mathbb{Z}_6$ acting on itself by multiplication, the sequences are $(3,2)$ and $(2,3),$ so the condition is satisfied.
Example 5: For the wreath product $S_n \wr S_m$ acting as usual on $[n] \times [m],$ the lattice of block systems is a chain of length $2.$ The only sequence is $(m, n),$ and hence the condition is satisfied.
This condition seems natural to me from the point of view of Ritt's polynomial decomposition theorems and related work. In that context, $G$ is a Galois group acting on the fibers of some polynomial or rational map of finite degree, and we want to "factor" that map as a composition of "irreducible" maps. The condition should then imply that the set of degrees of the irreducible elements are independent of the decomposition. For instance, corresponding to Example 4, the decompositions $(z^3)^2 = (z^2)^3$ have the same set of associated degrees.
Rather than characterizing such actions (although that would most certainly be welcome!), I am mostly just wondering if this condition or some close cousin already has an established name in the literature.
