Functoriality of group homology I understand that group homology $H_*(-)\colon \mathsf{Grps} \to \mathsf{Ab} $ is a functorial. In Weibel's homological algebra, there is an argument in 6.7 show this by using that $H_*(G;-)\colon G\mathsf{-mod} \to \mathsf{Ab}$ is a left derived functor and thus a universal $\delta$-functor.
For a given group homomorphism $\phi\colon H\to G$, I would really like to understand the induced map $H_*(\phi)$ better. Is there for example a chain map 
$$E_*H\otimes_{\mathbb ZH} \mathbb Z \to E_*G\otimes_{\mathbb ZG} \mathbb Z$$
for some free (or projective or flat) $\mathbb ZH$- and $\mathbb ZG$- resolutions $E_*H$ and $E_*G$ respectively of the trivial representation $\mathbb Z$?
In particular when $H\le G$, I thought the quotient map
$$ E_*G\otimes_{\mathbb ZH} \mathbb Z \to E_*G\otimes_{\mathbb ZG} \mathbb Z$$
would give such a chain map. If that was the case though, what does this mean for isomorphisms $\phi\colon G\to G$ and doesn't that induce the identity on
$$ E_*G\otimes_{\mathbb ZG} \mathbb Z \to E_*G\otimes_{\mathbb ZG} \mathbb Z$$
and thus the identity on the homology and not an isomorphism as I would expect unless it's an inner isomorphism.
Can you help me clear up this matter a bit?
 A: Let $\phi\colon H \to G$ be a group homomorphism, $M$ a $\mathbb ZH$-module, $N$ a $\mathbb ZG$-module, and $\psi\colon M \to \operatorname{Res}_\phi N$ an $H$-equivariant map. We want to understand the induced map
$$H_*(H;M) \longrightarrow H_*(G;N).$$
Let $E_*H$ and $E_*G$ be free $\mathbb ZH$- and $\mathbb ZG$-resolutions of the trivial representation $\mathbb Z$, respectively. Let $f\colon E_*H \to E_*G$ be an $H$-equivariant chain map that respects the augmentation. This exists by extending the identity map on $\mathbb Z$. This map is unique up to homotopy. It induces a map
$$ E_*H\otimes_H  M \stackrel{\phi_*\otimes \mathrm{id}}{\longrightarrow} E_*G\otimes_H  M \stackrel{\mathrm{id} \otimes \psi}{\longrightarrow} E_*G\otimes_H \operatorname{Res}_\phi N \twoheadrightarrow E_*G\otimes_G N.$$
Then this map induces a unique map on homology. 
More concretely let $E_*$ for example be the (unnormalized) bar resolution, i.e.
$$ E_n G = \bigoplus_{(g_1,\dots,g_n)\in G^n} \mathbb ZG$$
is a free right $\mathbb ZG$-module where we denote the generators by $[g_1\otimes \dots \otimes g_n]$. Then
$$[h_1\otimes \dots \otimes h_n]\cdot h_0 \mapsto [\phi(h_1)\otimes \dots \otimes \phi(h_n)] \cdot \phi(h_0)$$
gives an $H$-equivariant chain map $E_*H \to E_*G$ that respects the augmentation.
Now for the special case $H=G$, $M=N$, $\phi=c_g$ conjugation ($g'\mapsto gg'g^{-1}$) with $g\in G$, and $\psi\colon M\to \operatorname{Res}_\phi M$ is given by $m\mapsto gm$. We can simplify $\phi_*\colon E_*G \to E_*G$ setting it $x \mapsto xg^{-1}$. Then we get a chain map
$$E_*G \otimes_G M \to \operatorname{Res}_\phi E_*G \otimes_G \operatorname{Res}_\phi M = E_*G \otimes_G M$$
given by
$$ x\otimes m \longmapsto xg^{-1} \otimes gm  = x\otimes m.$$
Therefore conjugation induces the identity on homology (and even on the chain level, if setup right).
Reference: K. Brown: Cohomology of Groups, III.8
