MOD-2 (CO)homology of an Abelian Group

It is known that for a prime p ≠ = 2, there is the following natural description of the homology algebra of an Abelian group H_*(A, 𝔽_p) ≅ Λ(A/p)⊗Γ(_pA), and for finitely generated Abelian groups there is the following description of the cohomology algebra of H^*(A, 𝔽_p) ≅ Λ((A/p)^∨)⊗Sym((_pA)^∨). It is proved that for p = 2, there are no such descriptions “depending” on A/2 and ₂A only. Moreover, natural descriptions of H_*(A, 𝔽₂) and H^*(A, 𝔽₂), “depending” on A/2, ₂A, and a linear map $\overline{\beta}$: ₂A → A/2 are presented. It is also proved that there is a filtration by subfunctors on H_n(A, 𝔽₂), whose quotients are Λⁿ⁻²ⁱ(A/2)⊗Γⁱ(₂A), and there is a natural filtration on Hⁿ(A, 𝔽₂) for finitely generated Abelian groups, whose quotients are Λⁿ⁻²ⁱ((A/2)^∨) ⊗ Symⁱ((₂A)^∨).