Belyi injectivity for outer representations on certain quotients of étale fundamental groups of hyperbolic curves of genus zero

In the present paper, we study certain quotients of étale fundamental group a hyperbolic curve over field. We prove that action outer automorphism quotient an algebraically closed field on its conjugacy classes open subgroups is faithful. Also, that, if k either number or p-adic local field, then Galois representation associated to geometric ℙk1\{0,1,∞} injective.