Given a group $G$, a $\mathbb Z$-graded cohomology theory $E^*_G$, and a $n$-sheeted covering $p\colon X \to Y$, I would like a transfer map $$p_!\colon E^*_G X \to E^*_G Y$$ satisfying $$\require{cancel}\xcancel{p_! p^* = n \cdot \mathrm{id}.}$$ [this formulation was wrong; see the comments] such that when $n$ is inverted, $p_! p^*$ becomes an isomorphism (what I really need is that $p^*$ is an injection to a direct factor).

I'm aware such transfers exist for arbitrary fibrations $p$ (the number $n$ becomes the Euler characteristic of the fiber) under the additional assumption $E^*_G$ is $RO(G)$-graded, and that the $RO(G)$-grading is necessary for that result, but I only need them for covering maps, and I'd like to avoid the additional hypothesis.

So, are there transfer maps in this generality?

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    $\begingroup$ I thought that the equation $p_!p^\ast=n$ was false already when $G$ is trivial and $n=2$ (for generalized cohomology theories). $\endgroup$ – Tom Goodwillie Jul 7 '17 at 21:24
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    $\begingroup$ @user51223: I don't mean equivariantly, the equation is false for the ordinary transfer. The push-pull formula is true for any multiplicative cohomology theory, but shows that $t \circ p$ is multiplication by $t(1) \in \pi^0(Y)$, which is why stable cohomotopy plays a distinguished role. Mapped to ordinary cohomology this is a scalar, namely $\chi(F)$, but that is not generally true. In a general cohomology theory it is $\chi(F)$ modulo higher Atiyah-Hirzebruch filtration, and in particular becomes a unit if $\chi(F)$ is inverted (at least if the base of the fibration is a finite complex). $\endgroup$ – Oscar Randal-Williams Jul 8 '17 at 19:51
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    $\begingroup$ @user51223, I believe it's the second meant. The filtration $F_p E^*(Y) = \ker(E^* Y \to E^* Y^{p-1})$ gets you $E_2^{p,q} = H^p(Y;E^q(*))$. In the Becker–Gottlieb proof, the additional assumption that $\chi(F)$ is invertible makes $p_! p^*\colon E^* Y \to E^* Y$ induce an automorphism of this $E_2$ page, so that $p_! p^*$ is an automorphism. $\endgroup$ – jdc Jul 9 '17 at 20:56
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    $\begingroup$ If you ask for a structured enough version of these transfers, and G is finite, then I think this is equivalent to asking for an RO(G)-graded cohomology theory (basically by the equivariant version of Segal's machine.) If you drop down the structure, there might be counterexamples akin to the classical counterexamples of the "transfer conjecture" in nonequivariant homotopy theory. Basically, it's like measuring the difference between an H_infty-space and an E_infty space... there are examples of the former which aren't the latter, but they tend to be weird. In your case, it's like you have $\endgroup$ – Dylan Wilson Jul 26 '17 at 21:16
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    $\begingroup$ an E_infty G-space and you want to give it a "G-H_infty" structure without requiring that it be "G-E_infty". This is a little different than the classical transfer story (which begins with just an arbitrary H_infty space and asks whether it's also E_infty) since you do have non-equivariant coherent multiplication. But I think I would still be a little surprised if it turned out that coherent addition + H_infty-style-transfers = coherent transfers. $\endgroup$ – Dylan Wilson Jul 26 '17 at 21:19

Let me try to turn my comments into something like an answer (but the 'tldr' version is "I don't know.")

  1. When $G=*$, a space $X$ represents a functor with homotopy coherent transfers if and only if $X$ is a $\Gamma$-space (and hence equivalent to an $E_{\infty}$-space). Actually, you might take 'being a $\Gamma$-space' as the definition of having homotopy coherent transfers. One could (and Quillen did) ask whether this is equivalent to just asking that $[-,X]$ has functorial transfers for finite covers. That was called "the transfer conjecture". This turns out to be the same as asking that $X$ be an $H_{\infty}$-space. And so one asks "Is every $H_{\infty}$-space an $E_{\infty}$-space?" The answer is no, and a counterexample was provided by Kraines and Lada.
  2. For finite $G$, a $G$-space $X$ represents a functor with homotopy coherent transfers for finite covers (of $G$-spaces) if and only if it admits the structure of an equivariant Segal space (and hence is equivalence to a $G-E_{\infty}$-space. So, up to group completion, $X$ is the zeroth space of a genuine $G$-spectrum (and so represents an $RO(G)$-graded cohomology theory).
  3. If $X$ is a $G$-space representing a functor with homotopy coherent transfers for finite covers fibered in trivial $G$-sets, then this is like saying $X$ is an $E_{\infty}$-space in $G$-spaces. In particular, up to group completion, it admits ordinary deloopings and represents a $\mathbb{Z}$-graded cohomology theory for $G$-spaces.
  4. In your situation, you seem to have a space $X$ as in (3) and are asking if it's possibly to ask that $[-,X]$ admit transfers for all covers without requiring that $X$ admit $RO(G)$-deloopings, i.e. without requiring that $X$ be equivalent to a $G-E_{\infty}$-space. My guess is that this is possible, but that any example would be manufactured (like the counterexample of Kraines and Lada). However, I should point out that this situation is not precisely parallel to that in (1) because you have already placed an $E_{\infty}$-structure on $X$, so it's at least plausible that this homotopy coherence together with some weak notion of more exotic transfers could be enough... but again, I doubt it.
  5. In your last comment you mention that you are only interested in transfers for covers with cyclic structure group. Even for those, I think I stand by my intuition from (4), but I could be wrong. One imagines that whatever obstruction is responsible for establishing the conjectured example in (4) would be known to cyclic covers, especially after localizing at a prime.
  6. I've gone this whole answer without saying $N_{\infty}$-operad. Consider it said. (It's relevant to this business of asking for fewer transfers, somewhere between "fibered in trivial $G$-sets" and "fibered in arbitrary $G$-sets". Blumberg-Hill is the place to learn about these.)
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