I am trying to compute a Lyndon–Hochschild–Serre spectral sequence with coefficient $\mathbb{Z}/p^2\mathbb{Z}$, where $p$ is an odd prime, and Kudo's transgression theorem looks like a good tool for me to obtain the $E_\infty$-page. However, all the versions of Kudo's Theorem I can find are given under coefficient $\mathbb{F}_p$.
My question is: I understand that we can extend the coefficient to some field $K$ of characteristic $p$, but what if the coefficient is $\mathbb{Z}/p^2\mathbb{Z}$ or $\mathbb{Z}/p^n\mathbb{Z}$, where $p$ is an odd prime?
As a reference, here is a statement of the Kudo's transgression theorem provided by "A user's guide to spectral sequences", John McCleary, where the coefficient is $\mathbb{Z}/p\mathbb{Z}$:
Theorem If a class $x$ in $E^{0, 2k}_2\cong H^{2k}(F; \mathbb{F}_p)$ is transgressive and $x$ transgresses to the element represented by $y$ in $E^{2k+1, 0}_2\cong H^{2k+1}(B; \mathbb{F}_p)$, then $P^kx = x^p$ and $y\otimes x^{p-1}$ are also transgressive with $d_{2pk+1}(x^p) = P^ky$ and $d_{2(p-1)k+1}(y\otimes x^{p-1}) = -\beta P^k y$. (Here $\beta$ denotes the mod $p$ Bockstein in the Steenrod algebra $\mathcal{A}_p$.)
