In the arithmetic study of fundamental groups of algebraic curves over finite fields, the liftings of the Frobeniuses play an essential role. Let C be a smooth proper irreducible algebraic curve over a finite field F;I and let II (resp. n’) be the graphs on C x C of the qth (resp. q-‘th) power correspondences of C. In several interesting cases, C admits a lifting to characteristic 0 together with its correspondence T = 17 U P. It will be shown that in such a case the quotient of the algebraic fundamental group xi(C) modulo its normal subgroup generated by the Frobenius elements of “special points” (see below) is determined by the homomorphisms between the topological fundamental groups of the liftings of C and of T. The main result of this paper, which generalizes [2a, b] (Sect. 4), is formulated roughly as follows. Suppose that there exist two compact Riemann surfaces 31, 93’ that Yift” C, and another one, go, equipped with two finite morphisms p: 91°+ 93, o’: 93’ -+ 3’ such that 9 x @: ‘3’ --P 3 X 3’ is generically injective and (o X 4~‘) (‘3”) lifts T. Call an F,,rational point x of C special if the point (x, x9) of T (which lies on the crossing of 17 and n’) does not lift to a singular point of (u, x rp’)(l#‘). Then