Mod-2 (co)homology of an abelian group

It is known that for a prime $p\mathrm{\ne }2$ 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 $H^{\ast }(A,{\mathbb{F}}_p)\cong \mathrm{\Lambda }((A\mathrm{/}p{)}^{\vee })\otimes \mathsf{S}\mathsf{y}\mathsf{m}(({}_{p}A{)}^{\vee })\mathrm{.}$ 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 $A\mathrm{/}2,$ ${}_{2}A$ and a linear map $\tilde{\beta }:{}_{2}A\to A\mathrm{/}2.$ Moreover, we prove that there is a filtration by subfunctors on $H_n(A,{\mathbb{F}}_2)$ whose quotients are ${\mathrm{\Lambda }}^{n-2i}(A\mathrm{/}2)\otimes {\mathrm{\Gamma }}^i({}_{2}A)$ and that for finitely generated abelian groups there is a natural filtration on $H^n(A,{\mathbb{F}}_2)$ whose quotients are ${\mathrm{\Lambda }}^{n-2i}((A\mathrm{/}2{)}^{\vee })\otimes {\mathsf{S}\mathsf{y}\mathsf{m}}^i(({}_{2}A{)}^{\vee })\mathrm{.}$