Let $X$ be a connected space. According to Getzler BV-algebras and two-dimensional topologcial field theories , page 271, we have and isomorphism

$ H_*(\Omega^2\Sigma^2X) \cong {\cal G}( \widetilde{H}_* X ) $

where ${\cal G}( V)$ means the free Gerstenhaber (Getzler calls it "braid") algebra over the graded space $V$ and $\widetilde{H}$ is the reduced homology.

Getzler credits Cohen's results in The homology of iterated loop spaces for this isomorphism. The closest thing I can find there is Cohen's theorem 3.2, in his chapter "The homology of $C_{n+1}$-spaces, $n\geq 0$", which sounds like it, but I'm having some problems to deduce Getzler's claim.

First of all, Getzler says to be working with complex coefficients, and Cohen with $\mathbb{Z}_p$ ones. Is it clear that the result should be true no matter which coefficients? Rational coefficients too?

Secondly, Cohen's result for $n=1$ would be, I guess, Getzler's case:

$ H_*(\Omega^2\Sigma^2X) \cong GW_1(H_*X) \ . $

But here the free algebra functor is this $GW_1$ which I'm having some troubles to identify with ${\cal G}$.

Any hints or other references will be greatly appreciated.