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?