The following construction is taken from the book Central simple algebras and Galois cohomology by Gille and Szamuely in Chapter 6.4:
Let $C$ be a smooth projective curve over a perfect field $k$ with function field $K$. For an algebraic closure $\bar{k}$ of $k$, let $\bar{C} := C \times_k \bar{k}$ and let $G := \mathrm{Gal}(\bar{k}/k)$. The function field of $\bar{C}$ is by definition the composite $K\bar{k}$.
Let $C_0$ denote the closed points of $C$. For $P \in C_0$, $G$ permutes the closed points $Q$ in $\bar{C}_0$ lying over $P$, so we get a decomposition $$\mathrm{Div}(\bar{C}) = \bigoplus _{P \in C_0} \left(\bigoplus_{Q \mapsto P}\mathbb{Z} \right).$$ Therefore for each $i \geq 0$, the divisor map $(K\bar{k})^\times \rightarrow \mathrm{Div}(\bar{C})$ induces the map $$H^i(G,(K\bar{k})^\times) \rightarrow \bigoplus _{P \in C_0} H^i \left(G,\bigoplus_{Q \mapsto P}\mathbb{Z} \right) \cong \bigoplus _{P \in C_0}H^i(G_P,\mathbb{Z}),$$ where $G_P \subset G$ denotes the stabilizer of a fixed $Q$ in $\bar{C}$ and the last map follows from Shapiro's lemma. For $i \geq 2$, it is well-known that $H^i(G_P,\mathbb{Z}) \cong H^{i-1}(G_P,\mathbb{Q}/\mathbb{Z})$. For our case, we fix $i=2$ and a $P \in C_0$ to obtain the residue map $$r_P: H^2(G, (K\bar{k})^\times) \rightarrow H^1(G_P,\mathbb{Q}/\mathbb{Z}) = \mathrm{Hom}(G_P,\mathbb{Q}/\mathbb{Z}).$$ By our construction, we know that such maps are trivial for all but finitely many $P$.
Question. If $r_P$ is non-trivial, can we find an automorphism $\tau: C \rightarrow C$ and a $P' \in C_0$ such that $\tau(P) = P'$ and $r_{P'}$ is trivial?
Note that this is automatically true when $C$ is an elliptic curve. Also, for a curve of genus $\geq 2$, it is known that it has finitely many automorphisms, so if the answer to the question is affirmative, it would not be straightforward (to me).