Let $p$ be an odd prime and $n>1$. I am trying to understand why the cohomology ring $H^{\ast}(\mathbb{Z}/p^n;\mathbb{F_p})$ is given by $$\mathbb{F}_p[y]\otimes\Lambda(x),$$ with $|x|=1,|y|=2$ and $\beta_n(x)=y$, where $\beta_n:H^{1}(\mathbb{Z}/p^n;\mathbb{F_p})\to H^{2}(\mathbb{Z}/p^n;\mathbb{F_p})$ is the $n$th Bockstein map. I have seen a few abstract definitions of $\beta_n$ as a higher cohomology operation or as a higher differential in the Bockstein spectral sequence. The first questions that come to my mind are:

-Can we describe explicitly $\beta_n(x)=y$ as a $2$-cocycle?

-Is the group $E=\mathbb{Z}/p\times_{y}\mathbb{Z}/p^n$ determined by $y$ abelian?

Of course, the latter question would follow from the former.