Let $X$ be a smooth projective curve defined over a number field $K$. Let $\overline{K}$ denote the algebraic closure of $K$, and set $\overline{X} := X\otimes \overline{K}$. Denote by $\iota: \overline{X}\longrightarrow X$ the canonical morphism.

Let $\mathcal{F}$ denote an etale local system on $X$, and let $\iota^*\mathcal{F}$ denote its pullback to $\overline{X}$.

Is there a "Poitou-Tate duality"-type theorem, which gives an isomorphism between the $\mathbb{Q}_p$-vector groups:

$$\text{Ker}(H^1_{et}(X,\mathcal{F})\longrightarrow H^1_{et}(\overline{X},\iota^*\mathcal{F})),$$ and $$\text{Ker}(H^2_{et}(X,\mathcal{F})\longrightarrow H^2_{et}(\overline{X},\iota^*\mathcal{F})),$$ potentially replacing $\mathcal{F}$ by a twist of its dual if necessary?

Let $X$ be a smooth projective curve defined over a number field $K$. Let $\overline{K}$ denote the algebraic closure of $K$, and set $\overline{X} := X\otimes \overline{K}$. Denote by $\iota: \overline{X}\longrightarrow X$ the canonical morphism.

Let $\mathcal{F}$ denote an étale local system on $X$, and let $\iota^*\mathcal{F}$ denote its pullback to $\overline{X}$.

Is there a "Poitou–Tate duality"-type theorem, which gives an isomorphism between the $\mathbb{Q}_p$-vector groups:

$$\operatorname{Ker}(H^1_\text{ét}(X,\mathcal{F})\longrightarrow H^1_\text{ét}(\overline{X},\iota^*\mathcal{F})),$$ and $$\operatorname{Ker}(H^2_\text{ét}(X,\mathcal{F})\longrightarrow H^2_\text{ét}(\overline{X},\iota^*\mathcal{F})),$$ potentially replacing $\mathcal{F}$ by a twist of its dual if necessary?