Naive equivariant transfer

Question

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?

Answer 1

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.)