Some group-theoretic iteration technique is developed for an investigation of the generalized Abel functional equation \[ \alpha(f(x))= g(\alpha(x)). \] The technique is based on a change of a variable in intervals between adjacent fixed points of the given function \(f\). The change of variable together with a monotonicity property of \(f\) and \(g\) make it possible to introduce a group structure for the considered class of functions. The group structure is used in iterative constructing a solution \(\alpha\). This method is applicable provided \(f\) and \(g\) are both increasing or both decreasing.