Let $A$ be an abelian group and $p$ be a prime. If $p\ne 2,$ there is a very nice functorial description of the homology algebra $H_*(A,\mathbb Z/p):$ $$H_*(A,\mathbb Z/p)\cong \Lambda^*(A/p)\otimes \Gamma^{*/2}({}_pA),$$ where $\Lambda^*$ denotes the exterior algebra, $\Gamma^*$ denotes the divided power algebra and ${}_pA$ denotes $p$-torsion of $A$ (see Theorem 6.6 in Brown's "Cohomology of groups"). I like this isomorphism (like = it is useful for me) because

1. It is functorial.

2. The right part depends only on $A/p$ and ${}_pA$.

There is also such an isomorphism for $p=2$ but it is not functorial. And I do not like it because of this.

For any prime $p$ (including $p=2$) there is a short exact sequence of functors $$0\longrightarrow \Lambda^2(A/p) \longrightarrow H_2(A,\mathbb Z/p)\longrightarrow {}_pA \longrightarrow 0.$$ I like this short exact sequence because it gives a functorial 'description' of $H_2(A,\mathbb Z/2)$ in terms of $A/2$ and ${}_2A.$ The word 'description' here in a weak sense because it is not an isomorphism. But it is ok for me.

Question: Is there a functorial 'description' of $H_n(A,\mathbb Z/2)$ in terms of $A/2$ and ${}_2A$? Here the word 'description' can be in some weak sense.