As is commented by @Connor Malin, the action of the Steenrod algebra on $H^*B\mathbb{Z}/p$ is not faithful. Consider the case $p=2$. $Sq^3Sq^1$ acts trivially on $H^*(B \mathbb{Z}/2)$. As a matter of fact, the computation of the Hopf ring structure of $H_*K(\mathbb{Z}/p,*)$ shows that for no finite $n$, the action of the Steenrod algebra on $H^*((B\mathbb{Z}/p)^n)$ can be faithful. One can see this more easily using the last section of T.Kashiwabara Hopf Rings and Unstable Operations, JPAA 94 (1994) 183-193, or even from classical computations of Steenrod algebra.

As is commented by @Connor Malin, the action of the Steenrod algebra on $H^*B\mathbb{Z}/p$ is not faithful. Consider the case $p=2$. $Sq^3Sq^1$ acts trivially on $H^*(B \mathbb{Z}/2)$, since $Sq^{2n+1}x^{2m}=0$ by the Cartan formula. As a matter of fact, the computation of the Hopf ring structure of $H_*K(\mathbb{Z}/p,*)$ shows that for no finite $n$, the action of the Steenrod algebra on $H^*((B\mathbb{Z}/p)^n)$ can be faithful. One can see this more easily using the last section of T.Kashiwabara Hopf Rings and Unstable Operations, JPAA 94 (1994) 183-193, or even from classical computations of Steenrod algebra.

As is commented by @Connor Malin, the action of the Steenrod algebra on $H^*B\mathbb{Z}/p$ is not faithful. Consider the case $p=2$. $Sq^3Sq^1$ acts trivially on $H^*(B \mathbb{Z}/2)$. As a matter of fact, the computation of the Hopf ring structure of $H_*K(\mathbb{Z}/p,*)$ shows that for no finite $n$, the action of the Steenrod algebra on $H^*(K(\mathbb{Z}/p,n))$ can be faithful. One can see this more easily using the last section of T.Kashiwabara Hopf Rings and Unstable Operations, JPAA 94 (1994) 183-193, or even from classical computations of Steenrod algebra.

As is commented by @Connor Malin, the action of the Steenrod algebra on $H^*B\mathbb{Z}/p$ is not faithful. Consider the case $p=2$. $Sq^3Sq^1$ acts trivially on $H^*(B \mathbb{Z}/2)$. As a matter of fact, the computation of the Hopf ring structure of $H_*K(\mathbb{Z}/p,*)$ shows that for no finite $n$, the action of the Steenrod algebra on $H^*((B\mathbb{Z}/p)^n)$ can be faithful. One can see this more easily using the last section of T.Kashiwabara Hopf Rings and Unstable Operations, JPAA 94 (1994) 183-193, or even from classical computations of Steenrod algebra.