Is there any Kunneth formula for homology group with cofficient in an abelian group I am reading Hatcher Chapter V on spectral sequence. This is a paragraph after Theorem 5.3:

The Kunneth formula and the universal coefficient theorem then combine to give an isomorphism $$H_n(B\times F;G)\cong \bigoplus_p H_p(B;H_{n-p}(F;G))$$
  where $B$ is a simply connected space, $F$ is any topological space, and $G$ is an arbitrary abelian group.

I guess the logic would be
$$H_n(B\times F;G)\cong \bigoplus_{p} H_p(B;G)\otimes H_{n-p}(F;G)\cong  \bigoplus_p H_p(B;H_{n-p}(F;G))$$
But for the first isomorphism, as far as I know, it is only true when $G$ is a field then we can use Kunneth formula. 
Moreover, for the second isomorphism, by the universal coefficient theorem, there should be $\mathrm{Tor}$ part:
$$0 \to H_i(X; \mathbf{Z})\otimes A \, \overset{\mu}\to \, H_i(X;A) \to \operatorname{Tor}(H_{i-1}(X; \mathbf{Z}),A)\to 0.$$
The last term is zero if one of them is free, which is not generally true.
So I want to know what real argument we use here. I know both of theorems are pure homological algebra results, then probably we can adjust the condition properly so that we can get the desired result?
 A: It is convenient for an answer to this question   to generalise the usual homology  $H_*(C,A)$ of a chain complex $C$ (of abelian groups, and with $C_n= 0$ for $n 
< 0$) from  coefficients in an abelian group $A$,  to the case where $A$ is also a similar chain complex. We work with  the definition $H_*(C;A)=H_*(C \otimes A)$. 
So we are thinking of the questioner's  $H_n(B \times F;G)$ as $H_n(B
; C(F;G))$, where $C(F;G)$ is of course $C(F) \otimes G$, the chains of $F$ with coefficients in $G$.   
Then we use three  basic and well known properties of such chain complexes: 
1) for any  chain complexes   $F,A $ such that $F$ is free, and morphism $\phi: H_*(F) \to H_*(A)$ of graded groups, there is a morphism $f: F \to A$ of chain complexes such that $H_*(f)=\phi$; in  particular, if $F$ is free. there is a morphism $f: F \to H_*(F)$ such that $H_*(f)$ is the identity.
2) if $F$ is a free chain complex and $g: A \to B$ is a morphism of chain complexes such that $H_*(g): H_*(A) \to H_*(B) $ is an isomorphism, then
$1\otimes g: F \otimes A \to F \otimes B$ induces an isomorphism of homology; 
3) if $A$ is a chain complex there is a free chain complex $L$ such that there is a  morphism $a: L \to A$ inducing an isomorphism in homology. 
From this we deduce that if $A$ is a chain complex and $F$ is a free chain complex then there is an  isomorphism 
$$\kappa_F:H_*(F;A) \to H_*(F;H_*(A))$$
which can be chosen to be natural with respect to maps of $F$. To get $\kappa_F$ we choose a free chain complex $L$ and a morphism $a: L \to A$ inducing an isomorphism in homology. Then we choose a morphism $b: L \to H_*(L) $ inducing an isomorphism in homology. 
This can lead to specific calculations of $\kappa_F$. 
This is the dual of arguments for cohomology in this paper Chains as coeficients, (Proc. LMS  (3) 14 (1964) 545-65) and  examples are given  there of non  naturality.
The original  problem as suggested by M.G. Barratt was   to get  some results on Postnikov invariants of function spaces $X^Y$ by induction on the Postnikov system of $X$. 
