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Relative Rota--Baxter groups, a generalization of Rota--Baxter groups, areclosely connected to skew left braces, which play a fundamental role inunderstanding non-degenerate set-theoretical solutions to the Yang-Baxterequation. In this paper, we explore the inducibility problem for automorphismsof abelian extensions of relative Rota--Baxter groups. This problem isintricately linked to the recently introduced second cohomology of relativeRota--Baxter groups. Specifically, we prove a Wells-like exact sequence forabelian extensions of relative Rota--Baxter groups. The sequence establishes aconnection among the group of derivations, certain automorphism group, and thesecond cohomology of relative Rota--Baxter groups, thereby giving precisestructural relationships between these groups.

A Wells-like exact sequence for abelian extensions of relative Rota--Baxter groups