  Relative Rota--Baxter groups, a generalization of Rota--Baxter groups, are
closely connected to skew left braces, which play a fundamental role in
understanding non-degenerate set-theoretical solutions to the Yang-Baxter
equation. In this paper, we explore the inducibility problem for automorphisms
of abelian extensions of relative Rota--Baxter groups. This problem is
intricately linked to the recently introduced second cohomology of relative
Rota--Baxter groups. Specifically, we prove a Wells-like exact sequence for
abelian extensions of relative Rota--Baxter groups. The sequence establishes a
connection among the group of derivations, certain automorphism group, and the
second cohomology of relative Rota--Baxter groups, thereby giving precise
structural relationships between these groups.
