I know this may be a trivial question, but I can't find the answer on, for example, Milne's online notes and Danilov's Cohomology of Algebraic Varieties.
Suppose $K$ is a number field (say), $\overline K$ its algebraic closure, $G_K:=\text{Gal}(\overline K/K)$, $X$ is a variety over $K$, $\mathcal F$ is a locally constant (or constructible (?), don't know if it's true) abelian sheaf on $X_{\text{et}}$. It is said that for any $i\geqslant 0$, there is a continuous $G_K$ action on $H_{\text{et}}^i(X_{\overline K},\mathcal F)$.
My question is, how can one define this action? I found the answer that if $i=1$ and $\mathcal F$ is the constant sheaf associated to an abelian group $A$, then $H_{\text{et}}^1(X_{\overline K},\mathcal F)=\text{Hom}(\pi_1^{\text{et}}(X_{\overline K}),A)$, using the following exact sequence $$ 0\to\pi_1^{\text{et}}(X_{\overline K})\to \pi_1^{\text{et}}(X)\to\pi_1^{\text{et}}(\text{Spec }K)\cong G_K\to 0 $$ we get the Galois action. But what about $i\neq 1$ or $\mathcal F$ is not constant sheaf?
[EDIT] After viewing other questions on this site, I found that it is the proper base change theorem: if $f:X\to\text{Spec }K$ is proper, then there is a canonical isomorphism $H_\text{et}^i(X_{\overline K},\mathcal F_{\overline K})\xrightarrow\sim(R^if_*\mathcal F)_{\overline K}$, the latter is a $G_K$-module. (Complaint: this theorem is contained in notes I mentioned, but it relates etale cohomology to Galois representations is unmentioned, strange.)
But what about $X$ not proper, for example $X=Y_1(N)$ the open modular curve?
[EDIT2] Thanks for Roland's answer, but could anyone explain the simplest example: when $X=\text{Spec }k$, why does this definition of Galois action on $H_{\text{et}}^0$ coincides with the action given by the following category equivalence?
\begin{align*} \mathbf{Sh}\big((\text{Spec }k)_{\text{et}},\mathbf{Ab}\big) &\cong\{\text{discrete abelian group with continuous }G_k\text{-action}\} \\ \mathcal F&\mapsto\varinjlim_{k'/k\text{ finite separable extension}} \mathcal F(\text{Spec }k') \\ \big(\text{Spec }k'\mapsto A^{\text{Gal}(\overline k/k')}\big)&\leftarrow A \end{align*}
In fact, at first I thought I understand this category equivalence, but later I found that I didn't.