Finite-group families detecting primitivity of free-group tuples

Determine which families of finite groups d4a5 have the property that, for every free group F and every tuple h of elements of F, measure preservation of the map h_G for every Gind4a5 implies that h is primitive in F.

Background

For a free group F on d generators and a tuple h=(h_1,8a...,h_t) of elements of F, the paper associates to each finite group G a map h_G:Gd8a...Gt by evaluating the tuple under homomorphisms from F to G. If h is primitive8a..., meaning that it is contained in a free generating set of F, then h_G is measure-preserving for every finite group G.

The unresolved issue is to characterize the families of finite groups that detect primitivity through this measure-preservation property. The paper notes positive results for symmetric groups and certain families of finite soluble groups, as well as a positive result for rank-two free groups with the family of general linear groups, but it does not settle the question for arbitrary families and free groups.

References

Let F be a free group, and let h be a tuple of elements of F. For which families \mathcal F of finite groups does the following implication hold? [ h_G \text{ is measure-preserving for every }G\in\mathcal F \quad\Longrightarrow\quad h\text{ is primitive}. ]

— Average Numbers of Homomorphisms to Random Modules over Free Group Algebras  (2608.23273 - González et al., 24 Aug 2026) in Introduction, Question 1 (labelled "separablefamily")