Connected objects of $\textit{F\'Et}_{\Spec(k)}$ are of the form $\Spec(k') \to \Spec(k)$ with $k'/k$ a finite separable extension. Then $X_{\Spec{k'}}$ is connected, as the morphism $X_{\overline{k}} \to X_{\Spec(k')}$ is surjective and $X_{\overline{k}}$ is connected by assumption. Thus $\pi_1(X) \to \pi_1(\Spec(k))$ is surjective by Lemma \ref{lemma-functoriality-galois-surjective}. \medskip\noindent Before we go on, note that we may assume that $k$ is a perfect field. Namely, we have $\pi_1(X_{k^{perf}}) = \pi_1(X)$ and $\pi_1(\Spec(k^{perf})) = \pi_1(\Spec(k))$ by Lemma \ref{lemma-perfection}. \medskip\noindent It is clear that the composition of the functors $\textit{F\'Et}_{\Spec(k)} \to \textit{F\'Et}_X \to \textit{F\'Et}_{X_{\overline{k}}}$ sends objects to disjoint unions of copies of $X_{\Spec(\overline{k})}$. Therefore the composition $\pi_1(X_{\overline{k}}) \to \pi_1(X) \to \pi_1(\Spec(k))$ is the trivial homomorphism by Lemma \ref{lemma-composition-trivial}. \medskip\noindent Let $U \to X$ be a finite \'etale morphism with $U$ connected. Observe that $U \times_X X_{\overline{k}} = U_{\overline{k}}$. Suppose that $U_{\overline{k}} \to X_{\overline{k}}$ has a section $s : X_{\overline{k}} \to U_{\overline{k}}$. Then $s(X_{\overline{k}})$ is an open connected component of $U_{\overline{k}}$. For $\sigma \in \text{Gal}(\overline{k}/k)$ denote $s^\sigma$ the base change of $s$ by $\Spec(\sigma)$. Since $U_{\overline{k}} \to X_{\overline{k}}$ is finite \'etale it has only a finite number of sections. Thus $$ \overline{T} = \bigcup s^\sigma(X_{\overline{k}}) $$ is a finite union and we see that $\overline{T}$ is a $\text{Gal}(\overline{k}/k)$-stable open and closed subset. By Varieties, Lemma \ref{varieties-lemma-closed-fixed-by-Galois} we see that $\overline{T}$ is the inverse image of a closed subset $T \subset U$. Since $U_{\overline{k}} \to U$ is open (Morphisms, Lemma \ref{morphisms-lemma-scheme-over-field-universally-open}) we conclude that $T$ is open as well. As $U$ is connected we see that $T = U$. Hence $U_{\overline{k}}$ is a (finite) disjoint union of copies of $X_{\overline{k}}$. By Lemma \ref{lemma-functoriality-galois-normal} we conclude that the image of $\pi_1(X_{\overline{k}}) \to \pi_1(X)$ is normal. \medskip\noindent Let $V \to X_{\overline{k}}$ be a finite \'etale cover. Recall that $\overline{k}$ is the union of finite separable extensions of $k$. By Lemma \ref{lemma-limit} we find a finite separable extension $k'/k$ and a finite \'etale morphism $U \to X_{k'}$ such that $V = X_{\overline{k}} \times_{X_{k'}} U = U \times_{\Spec(k')} \Spec(\overline{k})$. Then the composition $U \to X_{k'} \to X$ is finite \'etale and $U \times_{\Spec(k)} \Spec(\overline{k})$ contains $V = U \times_{\Spec(k')} \Spec(\overline{k})$ as an open and closed subscheme. (Because $\Spec(\overline{k})$ is an open and closed subscheme of $\Spec(k') \times_{\Spec(k)} \Spec(\overline{k})$ via the multiplication map $k' \otimes_k \overline{k} \to \overline{k}$.) By Lemma \ref{lemma-functoriality-galois-injective} we conclude that $\pi_1(X_{\overline{k}}) \to \pi_1(X)$ is injective. \medskip\noindent Finally, we have to show that for any finite \'etale morphism $U \to X$ such that $U_{\overline{k}}$ is a disjoint union of copies of $X_{\overline{k}}$ there is a finite \'etale morphism $V \to \Spec(k)$ and a surjection $V \times_{\Spec(k)} X \to U$. See Lemma \ref{lemma-functoriality-galois-ses}. Arguing as above using Lemma \ref{lemma-limit} we find a finite separable extension $k'/k$ such that there is an isomorphism $U_{k'} \cong \coprod_{i = 1, \ldots, n} X_{k'}$. Thus setting $V = \coprod_{i = 1, \ldots, n} \Spec(k')$ we conclude.