Let $p$ be an odd prime. The $\mathbb F_p$ cohomology of the cyclic group of order $p$ is well-known: $\mathrm{H}^\bullet(C_p, \mathbb F_p) = \mathbb F_p[\xi,x]$ where $\xi$ has degree 1, $x$ has degree 2, and the Koszul signs are imposed (so that in particular $\xi^2 = 0$). As a module over the Steenrod algebra, the only interesting fact is that $x = \beta \xi$, where $\beta$ denotes the Bockstein. The rest of the Steenrod powers can be worked out by hand.

There are two groups of order $p^2$. The $\mathbb F_p$ cohomology of $C_p \times C_p$, including its Steenrod powers, is computable from the Kunneth formula. For the cyclic group $C_{p^2}$, you have to think slightly more, because there is a ring isomorphism $\mathrm{H}^\bullet(C_p, \mathbb F_p) \cong \mathrm{H}^\bullet(C_{p^2}, \mathbb F_p)$, but the Bockstein vanishes on $\mathrm{H}^\bullet(C_{p^2}, \mathbb F_p)$. Still I think the Steenrod algebra action is straightforward to write down.

I want to know about the groups of order $p^3$. The abelian ones are not too hard, I think, and there are two nonabelian groups. The one with exponent $p^2$ is traditionally denoted "$p^{1+2}_-$", and the one with exponent $p$ is traditionally denoted "$p^{1+2}_+$". I care more about the latter one, but I'm happy to hear answers about both. And right now I care most about the prime $p=3$.

The cohomology of these groups was computed in 1968 by Lewis in The Integral Cohomology Rings of Groups of Order $p^3$. Actually, as is clear from the title, Lewis computes the integral cohomology, from which the $\mathbb F_p$-cohomology can be read off using the universal coefficient theorem. For the case I care more about, Lewis finds that $\mathrm{H}^\bullet(p^{1+2}_+, \mathbb Z)$ has the following presentation. (I am quoting from Green, On the cohomology of the sporadic simple group $J_4$, 1993.) The generators are: $$ \begin{matrix} \text{name} & \text{degree} & \text{additive order} \\ \alpha_1, \alpha_2 & 2 & p \\ \nu_1, \nu_2 & 3 & p \\ \theta_j, 2 \leq j \leq p-2 & 2j & p \\ \kappa & 2p-2 & p \\ \zeta & 2p & p^2 \end{matrix}$$ (For the $p=3$ case that I care most about, there are no $\theta$s, since $2 \not\leq 3-2$.) A complete (possibly redundant) list of relations is: $$ \nu_i^2 = 0, \qquad \theta_i^2 = 0, \qquad \alpha_i \theta_j = \nu_i \theta_j = \theta_k \theta_j = \kappa \theta_j = 0$$ $$\alpha_1 \nu_2 = \alpha_2 \nu_1, \qquad \alpha_1 \alpha_2^p = \alpha_2 \alpha_1^p, \qquad \nu_1\alpha_2^p = \nu_2 \alpha_1^p,$$ $$ \alpha_i\kappa = -\alpha_i^p, \qquad \nu_i\kappa = -\alpha_i^{p-1}\nu_i,$$ $$ \kappa^2 = \alpha_1^{2p-2} - \alpha_1^{p-1}\alpha_2^{p-1} + \alpha_2^{2p-2}, $$ $$ \nu_1 \nu_2 = \begin{cases} \theta_3, & p > 3, \\ 3\zeta, & p = 3. \end{cases}$$ From this Green (ibid.), for example, writes down a PBW-type basis.

Question: What is the action of the Steenrod algebra been on $\mathrm{H}^\bullet(p^{1+2}_+, \mathbb F_p)$?

I'm not very good at Steenrod algebras. Does the ring structure on the $\mathbb Z$-cohomology suffice to determine the action? For instance, the additive structure of $\mathrm{H}^\bullet(G, \mathbb Z)$ already determines the Bockstein action on $\mathrm{H}^\bullet(G, \mathbb F_p)$. If there is a systematic way to do it, where can I learn to do the computations?