  Let $(X, D_{X})$ be an arbitrary pointed stable curve of topological type
$(g_{X}, n_{X})$ over an algebraically closed field of characteristic $p>0$. We
prove that the generalized Hasse-Witt invariants of prime-to-$p$ cyclic
admissible coverings of $(X, D_{X})$ attain maximum. As applications, we obtain
an anabelian formula for $(g_{X}, n_{X})$, and prove that the field structures
associated to inertia subgroups of marked points can be reconstructed
group-theoretically from open continuous homomorphisms of admissible
fundamental groups. Moreover, the formula for maximum generalized Hasse-Witt
invariants and the result concerning reconstructions of field structures play
important roles in the theory of moduli spaces of fundamental groups developed
by the author of the present paper.
