In group cohomology, for $H$ a finite-index subgroup of $G$ and $M$ a $G$-module, there is a transfer (or corestriction) map $Cor : H^\* (H;M) \to H^\*(G;M)$.


In homotopy theory, there is a transfer map for finite covering spaces $\bar{X} \to X$, and it exists for all coefficient systems on $X$. It is given on homology (say) by sending a small simplex in $X$ to its finitely-many lifts to $\bar{X}$. The group transfer is obtained from the topological one by the finite covering space $BH \to BG$.


There is a fancier transfer, due to Becker and Gottlieb, for any map $E \to B$ whose homotopy fibre is stably equivalent to a finite complex. Can this be extended to give a transfer map for coefficient systems on $B$?



Transfer homomorphisms with coefficients
I believe the answer is yes. The idea should be to generalize the construction of the group-theoretic transfer. I'll describe how I think this ought to go. I'm pretty sure what I'm describing is a known construction, but I haven't been able to dig up a reference which describes it.


Let $f:X\to Y$ be a map of spaces (CW-complexes), and assume (for simplicity) that both are path connected. Then there is an inclusion of topological groups $\phi: G\to H$, such that $B\phi: BG\to BH$ is equivalent to $f:X\to Y$.


A *local system* on $X$, in the most general possible context, is a spectrum $M$ equipped with a $G$-action. (I'm not doing anything fancy here; in particular, I'm not doing equivariant stable homotopy theory. Just spectra with a $G$-action; a $G$ map $f:M\to N$ of such things is a weak equivalence if it is a weak equivalence of the underlying spectra.) 


Let $S\_G$ be the category of $G$-spectra. It'll be important to note that this is a closed monoidal category: the smash product $X\wedge Y$ of two objects in $S\_G$ is defined in $S\_G$ (smash the spectra, and use the diagonal $G$-action), as well as a function object $\mathrm{Hom}(X,Y)$ (take the spectrum function object, with diagonal $G$-action).


The Becker Gottlieb transfer $f\_\!$ of $f$ associated to an $M\in S\_H$ should be a map $M\_{hH}\to M\_{hG}$ (where "$hG$" is "homotopy orbits"). So, if $M=S^0$ with trivial $H$-action, you get a map $S^0\_{hH}\to S^0\_{hG}$, which is a map $\Sigma^\infty Y\_+\to \Sigma^\infty X\_+$. You should interpret the map $\pi\_\* f\_\!$ as being a map
$$
H\_\*(Y,M) \to H\_\*(X,f^\*M),
$$
where these are "homology with coefficients in the local systems $M$ and $f^\*M$". There should be a similar way to get a cohomolgy transfer. 


Let $F=\Sigma^\infty(H/G)\_+$, as a spectrum with $H$-action. Note that the space $H/G$ is just the fiber of $f:X\to Y$. 


Here is a sequence of maps in $S\_H$:
$$
M\to \mathrm{Hom}(F,F\wedge M) \leftarrow \mathrm{Hom}(F,M)\wedge F \to \mathrm{Hom}(F,M)\wedge F\wedge F \to M\wedge F \approx M\wedge\_G H.
$$
The first is smashing with the identity map of $F$; the second comes from smashing a map $F\to M$ with a map $S^0\to F$; the third comes from the diagonal map $H/G\to H/G\times H/G$; the fourth is "evaluation".


If $F$ has the homotopy type of a *finite* CW-spectra (i.e., if $H/G$ is finitely dominated), then the backwards map in this sequence is a weak equivalence. In this case, we get a map $M\to M\wedge\_G H$ of $H$-spectra, and taking homotopy orbits gives $M\_{hH}\to M\_{hG}$.


If $Y$ is contractible, this amounts to a map 
$$ 
S^0 \to \mathrm{Hom}(F,F) \leftarrow \mathrm{Hom}(F, S^0)\wedge F \to \mathrm{Hom}(F,
 S^0)\wedge F\wedge F \to S^0\wedge F,
$$
which is to say, a map $S^0\to \Sigma^\infty X\_+$, and in cohomology this will send $1\in H^0(X)$ to $\chi(X)\in H^0(\mathrm{point})$. So apparently we recover the Becker-Gottlieb transfer.


If the spaces aren't connected, you can still do all this, but you need to allow $H$ and $G$ to be groupoids.


(I learned this way of thinking from some paper of John Klein about the "dualizing spectrum" of a toplogical group; I can't find one where he addresses the BG transfer this way.)


