TITLE: Transfer homomorphisms with coefficients
QUESTION [6 upvotes]: 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 $B$?

REPLY [3 votes]: I think what Charles has written up above is closely related to what Becker and Gottlieb did in
Becker, J. C.; Gottlieb, D. H.
Transfer maps for fibrations and duality. 
Compositio Math. 33 (1976), no. 2, 107–133. 
They work fiberwise, but that is equivalent to working equivariantly in the sense that a $G$-space $X$ gives rise to a fibration $X \times_G EG \to BG$ (Borel construction; this induces a Quillen equivalence).