Let $X$ be a scheme and let $\overline x$ be a geometric point of $X$. The Galois theory for schemes states that the category of finite étale covering of $X$ is equivalent to the category of finite $G$-sets, where $G = \pi_1(X, \overline x)$ denotes the étale fundamental group of $X$. In particular, the isomorphism classes of connected finite étale covering of $X$ are in one-to-one correspondence with the open subgroups of $G$, and it is called a Galois covering of $X$ if the corresponding subgroup is normal. I am wondering if there is an infinite Galois theory for schemes. To be specific, can we define an infinite Galois covering $Y \to X$ of $X$ such that the group $\mathop{\mathrm{Aut}}_X(Y)$ corresponds to a closed normal subgroup of $G$.
I came up with this question while I was reading Milne's book Étale Cohomology. In the proof of [Chap. VI, Cor. 1.4], Milne says that there exists a Hochschild-Serre spectral sequence \begin{equation*} E_2^{p,q} = H_{\mathrm{\acute et}}^p(G_k, H^q(X_{k^\mathrm{sep}}, \mathcal F)) \Longrightarrow E^{p+q} = H_{\mathrm{\acute et}}^{p+q}(X, \mathcal F), \end{equation*} where $X$ is a scheme of finite type over a field $k$, $\mathcal F$ is an abelian sheaf on $X_{\mathrm{\acute et}}$, $G_k = \mathop{\mathrm{Gal}}(k^{\mathrm{sep}} \vert k)$, and $X_{k^{\mathrm{sep}}} = X \times_k k^{\mathrm{sep}}$. So I wonder if we can define the infinite Galois coverings such that $X_{k^\mathrm{sep}}$ is an infinite Galois covering of $X$.
