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.

Your first map fits in an action of the Steenrod algebra.

In fact $H^*_{ét}(X;\mathbb{F}_2)$ is the homology of $C^*_{ét}(X;\mathbb{F}_2)=R\Gamma(X;\mathbb{F}_2)$, an element of the derived category of $\mathbb{F}_2$ which has the property of being an $E_\infty$-algebra.[1]

This means that not only it is a commutative algebra for the derived tensor product (thus generating a cup product on homology) but the commutative and associative relations can be upgraded with some explicit homotopies that satisfy additional relations (the precise definition is a bit of a pain to write down without the appropriate formalism, but it boils down to be "as commutative as you can hope to get"). 

In particular its homology $H^*_{ét}(X;\mathbb{F}_2)$ has more structure than just the cup product: it has an action of an algebra called the Dyer-Lashof algebra. The negative degree part of the Dyer-Lashof algebra is exactly the Steenrod algebra (negative degree because, as every good homotopy theorist, I always use homological grading). So, $H^*_{ét}(X;\mathbb{F}_2)$ inherits an action of the Steenrod algebra from the $E_\infty$-structure on the chain level.

Incidentally this is exactly the same mechanism underlying the action of the Steenrod algebra on the cohomology of spaces, and in general every time you have a well behaved notion of "derived global sections" you can get a Steenrod algebra action out of the deal.

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.

[1] The $E_\infty$-structure is due to the fact that $R\Gamma$ is a lax monoidal functor in the derived sense, being the right adjoint of the pullback functor which is symmetric monoidal, and so it preserves algebras for every operad.