Generators of the kernels of the symmetric automorphism maps

Determine explicit generating sets of the kernels of the homomorphisms \widehat{\calQ}_{k,d}: \SymAut(F_k) \rightarrow \SymAut(H_{k,d}) and \calQ_{k,d}: \SymOut(F_k) \rightarrow \SymOut(H_{k,d}), where H_{k,d} is the free product of k copies of \Z/d\Z and the maps are induced by the natural surjection F_k \twoheadrightarrow H_{k,d}. The goal is to identify concrete elements that generate \Ker(\widehat{\calQ}_{k,d}) and \Ker(\calQ_{k,d}) for given integers k,d \ge 2.

Background

The paper studies isotopy versus equivariant isotopy, with applications to symmetric automorphism groups of free products. For integers k,d > 1, the authors define H_{k,d} as the free product of k copies of \Z/d\Z and introduce the natural surjection F_k \to H_{k,d}, which induces maps on symmetric automorphism and symmetric outer automorphism groups: \widehat{\calQ}{k,d}: \SymAut(F_k) \to \SymAut(H{k,d}) and \calQ_{k,d}: \SymOut(F_k) \to \SymOut(H_{k,d}).

They note that, aside from a classical embedding result of Birman–Hilden, these kernels have not been systematically studied; while some elements in the kernels are obvious, a full description of generators is unknown. The paper proves that both \Ker(\widehat{\calQ}{k,d}) and \Ker(\calQ{k,d}) are not finitely generated when k \ge 3, addressing one finiteness aspect. The quoted passage highlights the remaining gap: identifying which elements generate these kernels. This problem is motivated by understanding the algebraic structure of these kernels and their connections to mapping class groups via branched coverings.