  In this paper, we study some group-theoretic constructions associated to
arithmetic fundamental groups of hyperbolic curves over finite fields. One of
the main results of this paper asserts that any Frobenius-preserving
isomorphism between the geometrically pro-$l$ fundamental groups of hyperbolic
curves with one given point removed induces an isomorphism between the
geometrically pro-$l$ fundamental groups of the hyperbolic curves obtained by
removing other points. Finally, we apply this result to obtain results
concerning certain cuspidalization problems for fundamental groups of (not
necessarily proper) hyperbolic curves over finite fields.
