On the Borel cohomology of free loop spaces

Let $X$ be a space and let $K = H^*(X; \boldsymbol F_p)$ where $p$ is an odd prime. We construct functors $\overline \Omega$ and $\ell$ which approximate cohomology of the free loop space $\Lambda X$ as follows: There are homomorphisms $\overline \Omega(K) \to H^*(\Lambda X; \boldsymbol F_p)$ and $\ell(K)\to H^*(E\boldsymbol T\times_T\Lambda X;\boldsymbol F_p)$. These are isomorphisms when $X$ is a product of Eilenberg-MacLane spaces of type $K(\boldsymbol F_p,n)$ for $n \geq 1$.