Isomorphism classes of étale $k$-algebras with dimension $n$ such that $A\otimes_k E$ is $E$-split étale and $H^1(\text{Gal}(E/k),S_n)$

Question

For groups $G$, $H$, let $H^1(G,H)$ be the set of $H$-orbits in $\text{Hom}_{\text{Grp}}(G,H)$ under the right $H$-action $(f\cdot a):g\mapsto a^{-1}f(g)a$. Let $G$ be the Galois group of a finite Galois extension $E/k$. Consider the set $X_n$ of isomorphism classes of étale $k$-algebras $A$ with dimension $n$ such that $A\otimes_k E$ is a split étale $E$-algebra. I would like to find a bijection between $X_n$ and $H^1(G,S_n)$.

By Grothendieck's Galois theory, any $k$-algebra isomorphism $\phi:A\to B$ between said étale algebras uniquely corresponds to its pullback $\phi^*:\text{Hom}_{k-\text{alg}}(B,E)\to\text{Hom}_{k-\text{alg}}(A,E)$ between finite $G$-sets. However, what exactly should be the map that maps the isomorphism class of $A$ to an orbit of $\text{Hom}_{\text{Grp}}(G,S_n)$? I struggle to proceed here.