This question is motivated by the following example : let $X$ be a variety over a field $k$, with algebraic closure $\bar{k}$. The Galois group $G_k:=\mathrm{Gal}(\bar{k}/k)$ acts on $X_{\bar{k}}:=X\times_k \bar{k}$ via the second factor, and for the projection $p:X_{\bar{k}}\to X$ and an étale sheaf $F$ on $X_{ét}$, the sheaf $p^\ast F$ on $X_{\bar{k}}$ has a $G_k$-equivariant structure and we have $\Gamma_X(F)=\Gamma(G_K,\Gamma_{X_{\bar{k}}}(p^\ast F))$. Actually we have more : if $F$ and $G$ are bounded complexes of étale sheaves of abelian groups on $X$, then $$R\Gamma(G_k,R\mathrm{Hom}_{X_{\bar{k}}}(p^\ast F,p^\ast G))=R\mathrm{Hom}_X(F,G)$$

I'm interested in potential generalizations of this, but I don't know very much about equivariant sheaves and toposes so hopefully this will not be too naive. Here are my questions:

What does it mean to be a $G_k$-equivariant sheaf in the above ? I've heard only about $G$-equivariant sheaves for $G$ a discrete group, but here $G_k$ is profinite.

Do we actually have an equivalence of categories between étale sheaves on $X$ and $G_k$-equivariant étale sheaves on $X_{\bar{k}}$ ?

Is there some sense in which $X=X_{\bar{k}}/G_k$ ?

Does it make sense to say that the étale topos of $X_{\bar{k}}$ is the "universal $G_k$-topos" over the étale topos of $X$ ? Is there a way to give a precise meaning to that ?

In general, if $\cal{T}$ is a topos and $G$ is a profinite group, does there exist a "universal $G$-topos" over $\cal{T}$ ? By that I mean a topos $\hat{\cal{T}}$ with an "action" of $G$ and a map $\pi:\hat{\cal{T}}\to \cal{T}$ such that $\Gamma_{\cal{T}}(F)=\Gamma(G,\Gamma_{\hat{\cal{T}}}(\pi^\ast F))$ for $F\in\cal{T}$ and $$R\Gamma(G,R\mathrm{Hom}_{\bar{\cal{T}}}(p^\ast F,p^\ast G))=R\mathrm{Hom}_{\cal{T}}(F,G)$$ for $F$ and $G$ in the "bounded derived category of abelian group objects" ?

Specifically, does it exist for $\cal{T}=\mathrm{Sh}((\mathrm{Spec}(\mathcal{O}_K))_{ét})$ the (small) étale topos of the ring of integers $\mathcal{O}_K$ in a global or local field $K$ and $G=\mathrm{Gal}(\bar{K}/K)$ ? Is that universal topos given by sheaves on a familiar site ?