Homology of metastable Eilenberg–MacLane spaces

Compute the \(\mathbb{F}_2\)-homology of the metastable Eilenberg–MacLane spaces \(\Omega^\infty\Sigma^n\mathbb{F}_2\).

Background

The paper proposes an internal Adams-type spectral sequence for the metastable category. A central unresolved input is the comonad governing mod-2 homology in $P_2\mathcal{S}_*$, which is closely related to the homology of metastable Eilenberg–MacLane spaces.

Determining these homology groups would help identify the algebraic structure of the proposed $E_2$-page and could lead to a computable Adams spectral sequence internal to the metastable category.