I have two questions.

$\bf 1.$ First, a reference request. Let $G\cong{\mathbb F}_p^r$ for some integer $r\geq 0$ and let $V=G^*={\rm Hom}(G,{\mathbb F}_p)$. Then $(H(G,{\mathbb F}_p),+,\cup )$ is a ring, $$H(G,{\mathbb F}_p)\cong\begin{cases}S(V)&p=2\\ \Lambda (V)\otimes S(V)&p>2\end{cases}.$$ Moreover, if $p=2$ then $V={\rm Hom}(G,{\mathbb F}_p)$ identifies as $H^1(G,{\mathbb F}_p)$; if $p>2$ then $V$ from $\Lambda (V)$ identifies as $H^1(G,{\mathbb F}_p)$, while $V$ from $S(V)$ identifies with the image of $V=H^1(G,{\mathbb F}_p)$ via the Bockstein boundary map $\beta :H^1(G,{\mathbb F}_p)\to H^2(G,{\mathbb F}_p)$, which happens to be injective.

An alternative description is $$H(G,{\mathbb F}_p)\cong\begin{cases}{\mathbb F}_2[x_1,\ldots,x_r]&p=2\\ \Lambda (x_1,\ldots,x_r)\otimes{\mathbb F}_p[y_1,\ldots,y_r]&p>2\end{cases},$$ where $x_1,\ldots,x_r$ are a basis of $V$ and $y_i=\beta (x_i)$.

These results are proved via Kunneth formula.

My question is where I can find these results so that I can quote them. I saw them in a paper and in a book, but with no reference given. It seems that people regard them as "common knowledge". In the paper I mentioned the authors simply said "Recall that...", as if everybody knows this, but some need to be reminded in case they forgot.

$\bf 2.$ The second question is whether there are explicit formulas for these isomorphisms in the literature.

If $p=2$, then the isomorphism ${\mathbb F}_2[x_1,\ldots,x_r]\to H(G,{\mathbb F}_2)$ is given by $x_{i_1}\cdots x_{i_n}\mapsto x_{i_1}\cup\cdots\cup x_{i_n}\in H^n(G,{\mathbb F}_2)$.

If $p>2$, then the isomorphism $\Lambda (x_1,\ldots,x_r)\otimes{\mathbb F}_p[y_1,\ldots,y_r]\to H(G,{\mathbb F}_p)$ is given by $x_{i_1}\wedge\cdots\wedge x_{i_s}\otimes y_{j_1}\cdots y_{j_t}\mapsto x_{i_1}\cup\cdots\cup x_{i_s}\cup y_{j_1}\cup\cdots\cup y_{j_t}\in H^{s+2t}(G,{\mathbb F}_p)$.

How about the reverse isomorphisms? Did anybody see anything published regarding this problem?

I did obtained explicit formulas for the reverse isomorphisms, where the elements of $H(G,{\mathbb F}_p)$ are written in terms of normalized cocycles. However, I don't know wether these results are new. Somebody might have thought about them before.