Your first map fits in an action of the Steenrod algebra. This is because $H^*_{ét}(X;\mathbb{F}_2)$ is the homology of a chain complex $C^*_{ét}(X;\mathbb{F}_2)=R\Gamma(X;\mathbb{F}_2)$ that has the structure of an $E_\infty$-algebra over $\mathbb{F}_2$ (being the derived global section of a sheaf of commutative rings), so it has an action of the Dyer-Lashof algebra. In particular it has an action of the negative degree part of the Dyer-Lashof algebra, which is nothing more than the Steenrod algebra. This is exactly the same mechanism underlying the action of the Steenrod algebra in the ordinary cohomology of spaces.
Ideally we would want $H^*_{ét}(X;\mathbb{F}_2)$ to be an unstable module over the Steenrod algebra (that is $Sq^ix=0$ for $|x|>i$) but I don't know if this is true.
NOTE: The existence of two "Bocksteins" means that the algebra acting naturally on $\bigoplus_n H^*_{ét}(X;\mathbb{F}_2(n))$ is presumably bigger than just the Steenrod algebra. I think it could be interesting to figure out exactly what algebra this is. Unfortunately I'm unaware of any work in this direction.