In this paper, we develop the theory of “cuspidalizations” of the étale fundamental group of a proper hyperbolic curve over a finite or nonarchimedean mixed-characteristic local field. The ultimate goal of this theory is the group-theoretic reconstruction of the étale fundamental group of an arbitrary open subscheme of the curve from the étale fundamental group of the full proper curve. We then apply this theory to show that a certain absolute $p$-adic version of the Grothendieck Conjecture holds for hyperbolic curves “of Belyi type”. This includes, in particular, affine hyperbolic curves over a nonarchimedean mixed-characteristic local field which are defined over a number field and isogenous to a hyperbolic curve of genus zero. Also, we apply this theory to prove the analogue for proper hyperbolic curves over finite fields of the version of the Grothendieck Conjecture that was shown in [Tama].