Let $X$ be a simply connected space. By $Q$ I denote $\Omega^{\infty}\Sigma^{\infty}$. Then $QX$ is an infinite loop space and the homology $H(QX)$ in $\mathbb{F}\_p$ is a Hopf algebra over the Dyer-Lashof algebra. 


Now there is a monomorphism $H(X) \rightarrow$ $H(QX)$ induced from $X \rightarrow QX$.


My question is if $H(QX)$ is generated over $H(X)$ in some sense.
Or is there some other relation between these two homologies?



Homology of infinite loop spaces $QX$
Let $X$ be a connected space, and let $\lbrace x\_\lambda\rbrace$ be a homogeneous basis for $H\_\ast(X;\mathbb{F}\_2)$. Then 
$$H\_\ast(QX;\mathbb{F}\_2) = \mathbb{F}\_2 [Q^I x\_{\lambda} \mid I\mbox{ admissible of excess }e(I)>\mathrm{dim}\,\, x\_\lambda ].$$ 


That is, the homology of $QX$ is a polynomial algebra with generators certain iterated Kudo-Araki operations on the basis of the homology of $X$. There is a similar result with coefficients mod $p$, $p$ an odd prime, involving Dyer-Lashof operations and the Bockstein operator. (The situation is reminiscent of Serre's Theorem on the cohomology of Eilenberg-Mac Lane spaces.)


The reference is (Section 5 of)


Dyer, Eldon; Lashof, R. K.
*Homology of iterated loop spaces*.
Amer. J. Math. 84 1962 35–88.


You will also find a nice discussion in 


Eccles, P. J. *Characteristic numbers of immersions and self-intersection manifolds*. Topology with applications (Szekszárd, 1993), 197–216, Bolyai Soc. Math. Stud., 4, János Bolyai Math. Soc., Budapest, 1995.


