Suppose $(G,X)$ is a Shimura data, and $E$ be its reflex field. In page 33 of this paper, it constructs an etale local system on the canonical model $\mathrm{Sh}(G,X)_{K,E}$ (variety over $E$) for any open compact neat $K\subset G(\mathbb A_f)$, using a $\mathbb Q$-representation of $G(\mathbb Q)$. But I don't understand why the local system descends to $E$.
Roughly speaking, suppose $K'\subset K$ is a normal subgroup of finite index, then we know $\mathrm {Sh}(G,X)_{K'}$ is a Galois cover of $\mathrm {Sh}(G,X)_{K}$ with Galois group $K/K'$. If $T$ is a finite set with $K/K'$-action, then it defines a local system on $\mathrm {Sh}(G,X)_{K}$. Then the paper says "Since $\mathrm {Sh}(G,X)_{K'}\to \mathrm {Sh}(G,X)_{K}$ descends to $E$, the local system also descends to $\mathrm{Sh}(G,X)_{K,E}$ (the canonical model over $E$)". I don't know why, because the Galois group of $\mathrm {Sh}(G,X)_{K',E}$ over $\mathrm {Sh}(G,X)_{K,E}$ may be larger than $K/K'$, and we need a natural extension of the action of $K/K'$ on $T$ and I can't see how.
