Galois coinvariants and fundamental groups of arithmetic schemes

We study Galois coinvariants of the geometric étale fundamental group and étale cohomology of varieties over a finitely generated field $k$. We establish finiteness properties of these coinvariants for (possibly infinite) Galois extensions $L\mathrm{/}k$ such that $\mathrm{Gal}(L\mathrm{/}k{)}^{\mathrm{a}\mathrm{b}}$ has finite exponent. As an application, we obtain a finiteness result for étale fundamental groups of arithmetic schemes over number fields.