Composantes irr\'eductibles de lieux sp\'eciaux d'espaces de modules de courbes, action galoisienne en genre quelconque

In this paper we characterise the action of the absolute Galois group on the geometric finite cyclic groups without 'etale factorization of stack inertia of the profinite geometric fundamental group of moduli spaces of marked curves. As a complementary result, we give the same action on prime order profinite elements in genus 2.