OK, I finally get it. (Or I think so.) We have a pairing $\rho :H^n(G,{\mathbb F}_p)\times H_n(G,{\mathbb F}_p)\to {\mathbb F}_p$. Both $H^n(G,{\mathbb F}_p)$ and $H^n(G,{\mathbb F}_p)$ have dimension $N=\binom{n+r-1}r$. I want to write $[a]\in H^n(G,{\mathbb F}_p)$ in the basis $\alpha_1,\ldots,\alpha_N$ of $\in H^n(G,{\mathbb F}_p)$, which is made of $\tau (x_1^{(n_1)}\cdots x_r^{(n_r)})$ with $n_1+\cdots +n_r=n$. If $\rho$ is non-degenerate and $\beta_1,\ldots,\beta_N$ is the dual basis of $H_n(G,{\mathbb F}_p)$ then $[a]=\sum_kc_k\alpha_k$, with $c_k=\rho ([a],\beta_k)$. This might work.

Unfortunately, you are talking about things I'm not familiar with. I never worked with $H_*$ and at this moment I can't even state its definition. I too noticed the same action of the symmetric group and the shuffle permutations, which appear in my formulas. Very likely it's not a mere coincidence. If there is some duality between $H^*$ and $H_*$, then this might be used to determine my coefficients $c_{n_1,\ldots,n_r}$. In my paper I only use the basic properties of $H^*$ and the existence of the isomorphism whose inverse I want to find. (Corollary II.4.3 and Theorem II.4.4 in Adem--Milgram.)

Maybe he was a little hasty. Later he wrote a comment, but it disappeared, I guess he deleted it. In Brown V.5.3, which he suggested I should look at, $G$ is cyclic. I need $G={\mathbb F}_p^r$. Even if $r=1$, when $G$ is cyclic, V.5.3 only produces a reverse isomorphism if we describe the elements of $H^*(G,{\mathbb F}_p)$ in terms of that special resolution of period 2 for the cyclic groups. I need it in terms of the normalized bar resolution. To go from one to the other one needs an explicit homotopy, which is technically difficult. Even if you mange to do this, how about the case $r>1$?

Sorry, I meant ${\mathbb F}_p^r$. Forgot to add the exponent $r$. I'll try to read more carefully that section in Brown to see if it leads to explicit closed formulas for the isomorphisms. I'm using a different approach.

It does answer the first question. However, for the second question, Section V.5.3 in Brown doesn't deal with the case $G\cong{\mathbb F}_p$, but with the case when $G$ is cyclic. So I guess I should go ahead with writing a small paper on the explicit formula for the isomorphism. (Provided I don't find out that the result already exists. Maybe somebody saw it somewhere.) I'm not, by far, an expert in cohomology. I only took a one semester reading course from Brown's book about 20 years ago and then forgot everything. Only recently I needed some cohomology in my work.

Thank you for the answer. Corollary II.4.3 and Theorem II.4.4 in Adem--Milgram is exactly what I needed. Funny thing, this is the book I was mentioning for having the result without a reference. I saw the result for $p>3$ at the beginning of section III.3.