  The notion of Rota-Baxter groups was recently introduced by Guo, Lang and
Sheng [{\em Adv. Math.} 387 (2021), 107834, 34 pp.] in the geometric study of
Rota-Baxter Lie algebras. They are closely related to skew braces as observed
by Bardakov and Gubarev. In this paper, we study extensions of Rota-Baxter
groups by constructing suitable cohomology theories. Among others, we find
relations with the extensions of skew braces. Given an extension of Rota-Baxter
groups, we also construct a short exact sequence connecting various
automorphism groups, which generalizes the Wells short exact sequence.
