Motivated by the Benjamini-Schramm non-unicity of percolation conjecture we study the following question. For a given finitely generated nonamenable group Gamma, does there exist a generating set S such that the Cayley graph (Gamma, S), without loops and multiple edges, has non-unique percolation, i.e., p(c)(Gamma, S) < p(u) (Gamma,S)? We show that this is true if Gamma contains an infinite normal subgroup N such that Gamma/N is non-amenable. Moreover for any finitely generated group G containing Gamma there exists a generating set S' of G such that p(c)(G, S') < p(u) (G, S'). In particular this applies to free Burnside groups B(n,p) with n >= 2, p >= 665. We also explore how various non-amenability numerics, such as the isoperimetric constant and the spectral radius, behave on various growing generating sets in the group.