Other representations with single-label robustness behavior

Determine whether transitive permutation-group representations beyond regular and Frobenius representations, or all transitive representations of additional group families, have the property that removing a single derangement label doubles the independence number and that further label removals exhibit analogous robustness behavior.

Background

The paper establishes this behavior for regular and Frobenius representations and for transitive representations of abelian and generalized dicyclic groups. It asks whether comparable behavior occurs in other representation classes or across further families of groups.

References

Question 7.1. Are there other transitive group representations that have similar properties-either with respect to a class of representations (as for regular and Frobenius representations) or with respect to all transitive representations of some family of groups (like the generalised dihedral groups over abelian groups that include odd permutations)?

— A new measure of robustness of Erdős--Ko--Rado Theorems on permutation groups  (2502.14582 - Gunderson et al., 20 Feb 2025) in Question 7.1, Section 7