Abstract: Weyl groups for the Cuntz algebras were introduced implicitly by Cuntz at the end of the seventies. However, they remained largely ignored probably because of computational difficulties until about 30 years later, when the breakthrough work by Conti and Szymanski allowed to determine explicitly a huge number of their elements by a sophisticated condition with a clear combinatorial flavour. In recent times, Brenti, Conti and Nenashev pushed the boundaries of the involved combinatorial structures, obtaining for instance the first enumerative results (at the moment, only for cycles). In this talk I will report on recent work joint with F. Brenti and G. Nenashev (in preparation) where, building on the combinatorial machinery developed over the last few years, we construct certain subgroups of outer automorphisms of O_n. In particular, we are able to describe in detail the 46 distinct finite groups of outer automorphisms of O_4 lying in the outer reduced Weyl group and maximal at level 2, which were first determined by Szymanski and collaborators by clever computer-assisted methods. The notion of bicompatible subgroup of the permutations of a square grid will play a role in the discussion. The Operator Algebra Seminar schedule is here: https://sites.google.com/view/oastorvergata/home-page