Let $TOP$ be the stable homeomorphism space, with $TOP(n) = Homeomorphisms(\mathbb{R}^n)$. What is known about its $\mathbb{Z}_2$-homology $H_*(TOP, \mathbb{Z}_2)$? In particular I am interested in the map $H_*(TOP, \mathbb{Z}_2) \rightarrow H_*(G, \mathbb{Z}_2)$, where $G$ is the stable space of homotopy automorphisms of spheres.

Can $H_*(TOP, \mathbb{Z}_2)$ be computed knowing $H_*(G, \mathbb{Z}_2)$, $H_*(G/TOP, \mathbb{Z}_2)$ or $H_*(BTOP, \mathbb{Z}_2)$?

My specific question: Is the map $H_{4k+2}(TOP, \mathbb{Z}_2) \rightarrow H_{4k+2}(G, \mathbb{Z}_2)$ injective on the image of $\pi_{4k+2}(TOP)$ under the Hurewicz homomorphism? So for example, is it injective on primitives? Does somebody have an idea or a reference I could look at?

Even more specific (and also sufficient): Let $\kappa \in \pi_{14}^s = \pi_{14}G$ be the element with Kervaire invariant 0. It lifts to $\pi_{14}(TOP)$. Is its image under $h : \pi_{14}(TOP) \rightarrow H_{14}(TOP, \mathbb Z_2)$ zero? (By Lemma 4.3 of arxiv.org/abs/1504.06752v2, $h(\kappa)=0$ in $H_{14}G$.)