I only gather some info of special cases. There is always a natural (functorial) map $\bigwedge^\ast(A/2) \to H_\ast(A,\mathbb Z/2)$ and it is actually injective. Furthermore it's a natural isomorphism when ${}_2A=0$. But you must know this already, it's Theorem 6.4 of Brown's book.

Some necessary material is found in the following papers (in French) of Cartan, which Ken Brown references precisely at the end of Theorem 6.6. I'm not completely sure how much it gets you, I am not the best at French translations, but it does at the least mention a description when ${}_2A=A$.
http://www.numdam.org/numdam-bin/browse?id=SHC_1954-1955__7_1
Namely, the two papers which concern $p$ odd and $p=2$ respectively:
Détermination des algèbres $H_*(\pi, n; Z_p)$ et $H^*(\pi, n ;Z_p)$, $p$ premier impair
Détermination des algèbres $H_*(\pi, n; Z_2)$ et $H^*(\pi, n; Z_2)$ ; groupes stables modulo $p$

I only gather answers for special cases. There is always a natural (functorial) injection $\bigwedge^\ast(A/2) \to H_\ast(A,\mathbb Z/2)$, and furthermore it's a natural isomorphism if ${}_2A=0$. But you must know this already, it's Theorem 6.4 of Brown's book.

Some necessary material is found in the following papers (in French) of Cartan, which Ken Brown references precisely at the end of Theorem 6.6. I'm not completely sure how much it gets you, I am not the best at French translations, but it does at the least mention a description when ${}_2A=A$.
http://www.numdam.org/numdam-bin/browse?id=SHC_1954-1955__7_1
Namely, the two papers which concern $p$ odd and $p=2$ respectively:
Détermination des algèbres $H_*(\pi, n; Z_p)$ et $H^*(\pi, n ;Z_p)$, $p$ premier impair
Détermination des algèbres $H_*(\pi, n; Z_2)$ et $H^*(\pi, n; Z_2)$ ; groupes stables modulo $p$

The necessary material is found in the following papers (in French) of Cartan, which Ken Brown references precisely at the end of Theorem 6.6. I'm not sure if this fits your desires, because I don't quite understand what you mean and also I am not the best at French translations:

http://www.numdam.org/numdam-bin/browse?id=SHC_1954-1955__7_1

Namely, the two papers which concern $p$ odd and $p=2$ respectively:
Détermination des algèbres $H_*(\pi, n; Z_p)$ et $H^*(\pi, n ;Z_p)$, $p$ premier impair
Détermination des algèbres $H_*(\pi, n; Z_2)$ et $H^*(\pi, n; Z_2)$ ; groupes stables modulo $p$

I only gather some info of special cases. There is always a natural (functorial) map $\bigwedge^\ast(A/2) \to H_\ast(A,\mathbb Z/2)$ and it is actually injective. Furthermore it's a natural isomorphism when ${}_2A=0$. But you must know this already, it's Theorem 6.4 of Brown's book.

Some necessary material is found in the following papers (in French) of Cartan, which Ken Brown references precisely at the end of Theorem 6.6. I'm not completely sure how much it gets you, I am not the best at French translations, but it does at the least mention a description when ${}_2A=A$.
http://www.numdam.org/numdam-bin/browse?id=SHC_1954-1955__7_1
Namely, the two papers which concern $p$ odd and $p=2$ respectively:
Détermination des algèbres $H_*(\pi, n; Z_p)$ et $H^*(\pi, n ;Z_p)$, $p$ premier impair
Détermination des algèbres $H_*(\pi, n; Z_2)$ et $H^*(\pi, n; Z_2)$ ; groupes stables modulo $p$