We consider cohomology sets and exact sequences of groups with involutions. In particular, we study congruence subgroups of type $\Gamma_0(m)$ which are acted by the group generated either by the map $z \mapsto (-1/mz)$ of the upper half plane or by the map $x \mapsto (1/mx)$ of the set of irrational real numbers.
Takashi Ono. "On certain exact sequences for $\Gamma _0(m)$." Proc. Japan Acad. Ser. A Math. Sci. 78 (6) 83 - 86, June 2002. https://doi.org/10.3792/pjaa.78.83