Mod-$2$ (co)homology of an abelian group

It is known that for a prime $p\ne 2$ there is the following natural description of the homology algebra of an abelian group $H_*(A,\mathbb F_p)\cong \Lambda(A/p)\otimes \Gamma({}_pA)$ and for finitely generated abelian groups there is the following description of the cohomology algebra of $H^*(A,\mathbb F_p)\cong \Lambda((A/p)^\vee)\otimes \mathsf{Sym}(({}_pA)^\vee).$ We prove that there are no such descriptions for $p=2$ that “depend” only on $A/2$ and ${}_2A$ but we provide natural descriptions of $H_*(A,\mathbb F_2)$ and $H^*(A,\mathbb F_2)$ that “depend” on $A/2,$ ${}_2A$ and a linear map $\widetilde \beta\colon {}_2A\to A/2.$ Moreover, we prove that there is a filtration by subfunctors on $H_n(A,\mathbb F_2)$ whose quotients are $\Lambda^{n-2i}(A/2)\otimes \Gamma^i({}_2A)$ and that for finitely generated abelian groups there is a natural filtration on $H^n(A,\mathbb F_2)$ whose quotients are $ \Lambda^{n-2i}((A/2)^\vee)\otimes \mathsf{Sym}^i(({}_2A)^\vee).$