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.)

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).