This paper investigates a presolving method for handling symmetries in mixed-binary programs, based on inequalities computed from so-called Schreier-Sims tables. We show that an iterative application of this method together with merging variables will produce an instance for which the symmetry group is trivial. We then prove that the problem structure can be preserved for certain monotone binary problems, e.g., packing and covering integer programs, maximum stable set (or clique) problems and set covering. The result is then a changed instance of the same type such that existing black-box solvers can directly be applied. We showcase our approach on stable set problems via computational experiments. For a mixed set of graphs, the presolving method significantly outperforms a black-box branch-and-cut method, while solving the same number of instances as when applying state-of-the-art symmetry handling methods. For Johnson and coding theory graphs, we significantly outperform the stand-alone solver with and without its integrated symmetry handling. A similar speed-up can be observed when running the maximum clique solver CliSAT on the Johnson graphs.