Let $f: Y\longrightarrow X$ be a smooth proper map of smooth proper schemes over $\mathbb{Q}$, and let $\mathcal{F} = R^1_\text{ét}\overline{f}_*\mathbb{Q}_p$ denote the derived pushforward of $\overline{f}$, i.e. the sheaf with stalks $\mathcal{F}_x := H^1_\text{ét}(Y_x\otimes\overline{\mathbb{Q}}, \mathbb{Q}_p)$.
The local system $\mathcal{F}$ is smooth, and in particular flat, equipped with a $G_{\mathbb{Q}}$-action, which is unramified outside a finite set of primes $S$. Let $G_S$ denote the Galois group of the maximal extension of $\mathbb{Q}$ which is unramified outside $S$, and let $\mathcal{G}$ denote the sheaf $H^1(G_S, \mathcal{F})$, whose value on a $G_S$-stable open $U$ is the group cohomology $H^1(G_S, \mathcal{F}(U))$ (is it a sheaf? I am fine with sheafifying if it isn't).
Question: is $\mathcal{G}$ flat? If not, are there reasonable assumptions one could add which would make it flat? Without assuming its action on the stalks is isomorphic everywhere.