For $X$ a topological space, from the short exact sequence

$$ 0 \rightarrow \mathbb{Z}/2 \rightarrow \mathbb{Z}/4 \rightarrow \mathbb{Z}/2 \rightarrow 0 $$

we get a Bockstein homomorphism

$$H^i(X, \mathbb{Z}/2) \rightarrow H^{i+1}(X, \mathbb{Z}/2)$$

This is also known as the Steenrod square $Sq^1$.

Now suppose instead that $X$ is a variety over a (not algebraically closed) field. We still get a sequence

$$ 0 \rightarrow \mathbb{Z}/2 \rightarrow \mathbb{Z}/4 \rightarrow \mathbb{Z}/2 \rightarrow 0 $$

inducing a Bockstein homomorphism in **etale cohomology**

$$H^i_{et}(X, \mathbb{Z}/2) \rightarrow H^{i+1}_{et}(X, \mathbb{Z}/2).$$

However, there is *also* a short exact sequence

$$ 0 \rightarrow \mathbb{Z}/2 \rightarrow \mathbb{Z}/4(1) \rightarrow \mathbb{Z}/2 \rightarrow 0 $$

where $\mathbb{Z}/4(1)$ denotes the Tate twist, probably less confusingly written as $\mu_4$. This *also* induces a [presumably different] Bockstein map in etale cohomology

$$H^i_{et}(X, \mathbb{Z}/2) \rightarrow H^{i+1}_{et}(X, \mathbb{Z}/2).$$

**Question**: Which of these is the "right" Bockstein homomorphism?

This question is a little open-ended. My main criterion for "right" is that the map be "$Sq^1$ on etale cohomology", meaning it fits into an action of the Steenrod algebra.

There are other possible criteria. For instance, a literature search revealed that people have defined notions of "Bockstein homomorphism" and "Steenrod operations" on Chow rings, motivic cohomology, ... so "right" could also mean "compatible with these other things". (Hopefully the answer is the same.)

Relevant literature:

- Jardine's paper https://link.springer.com/chapter/10.1007%2F978-94-009-2399-7_5 gives a version with $\mathbb{Z}/4$
- Guillou and Weibeil (http://arxiv.org/pdf/1301.0872.pdf) describe a $Sq^1 : H^k(\mu_2^i) \rightarrow H^{k+1}(\mu_2^{2i})$, so the weight has doubled. I don't know if it fits into a Bockstein story.

Unfortunately, I can't really parse what's happening in these papers.

Some motivation/example: For $X \subset Y$ a closed embedding of smooth varieties of codimension $r$, I have a cycle class $[X] \in H^{2r}_X(Y; \mathbb{Z}/2)$. I would like to show that $Sq^1 [X]=0$ (it may not be true). Since the cycle class lifts to $H^{2r}_X(Y;\mathbb{Z}_{2}(r))$, this depends on which sequence is related to $Sq^1$.