Let $G= C_p \times C_p$, where the cyclic group in the first factor is generated by $a$ and the second factor by $b.$ Let $H_j$ be a subgroup of $G$ generated by $(a, b^j)$ where $0 \le j \le p-1.$ Consider the quotient map $\pi_j : G \to G/H_j$.
Then in cohomology it induces a ring map $\pi_j^\ast: H^\ast(G/H_j; \mathbb{F}_p) \to H^\ast(G; \mathbb{F}_p) $ i.e.,
$\pi_j^\ast: \mathbb{F}_p[x_j, \beta(x_j)] \to \mathbb{F}_p[y, z, \beta(y), \beta(z)]$ where $\beta$ is the respective Bockstein homomorphism.
$\mathbf{ Question.}$ What is $\pi_j^\ast(x_j)=?$
Thank you in advance. Any help would be appreciated.