Following Wikipedia (https://en.wikipedia.org/wiki/Postnikov_square), a Postnikov square is a certain cohomology operation from a first cohomology group $H^1$ to a third cohomology group $H^3$, introduced by Postnikov (1949). Eilenberg (1952) described a generalization taking classes in $H^t$ to $H^{2t+1}$.

I checked Eilenberg's paper, I find that the generalized Postnikov square requires that $t$ is odd.

In Browder and Thomas's paper: Axioms for the generalized Pontryagin cohomology operations (https://www.maths.ed.ac.uk/~v1ranick/papers/browthom.pdf), they also defined a generalized Postnikov square $$\mathfrak{P}:H^{2n}(-,\mathbb{Z}/2^r)\to H^{4n+1}(-,\mathbb{Z}/{2^{r+1}}).$$

My question: Is there a generalized Postnikov square $$\mathfrak{P}:H^t(-,\mathbb{Z}/p^r)\to H^{2t+1}(-,\mathbb{Z}/{p^{r+1}})$$ for even $t$ and odd prime $p$?

Thank you!

onlycohomology operations $H^t \to H^{2t+1}$ are of the form $u \mapsto k \beta'(u^2)$ for some scalar $k$, where $\beta'$ is the Bockstein associated to $0 \to \Bbb Z/p^{r+1} \to \Bbb Z/p^{2r+1} \to \Bbb Z/p^r \to 0$. What do you need from the Postnikov square? – Tyler Lawson Oct 26 at 10:37