= Solution
For the <K-theory transfer of a finite covering>, let $V\to Y$ be a <vector bundle> and define
$$
(p_!V)_x=\bigoplus_{y\in p^{-1}(x)}V_y.
$$
On an evenly covered neighborhood the $n$ sheets identify this with a <direct sum> of $n$ local <vector bundles>. On overlaps the identifications permute the sheets and apply the transition maps of $V$, so these local descriptions glue to a <vector bundle> on $X$. This operation respects <direct sums>, hence extends to a homomorphism of <Grothendieck groups>
$$
\boxed{p_!:K^0(Y)\longrightarrow K^0(X),\qquad
p_!([V]-[W])=[p_!V]-[p_!W].}
$$
It obeys the <projection formula for the K-theory transfer>:
$$
p_!(p^*a\cdot b)=a\cdot p_!b.
$$
For actual <vector bundles>, this follows fiberwise by distributing the <tensor product of vector bundles> over the <direct sum>; it then extends to their <Grothendieck groups>.
Let $P=p_!(1_Y)$ be the permutation <vector bundle> of the covering. It has rank $n$ on every <connected component>, but need not be a <trivial vector bundle>; in particular one cannot replace $p_!p^*$ by multiplication by $n$ integrally. Instead write
$$
[P]=n+\eta,\qquad \eta\in I(X).
$$
The componentwise nilpotence result from the preceding solution gives $\eta^N=0$ for some $N$. In the <localization of a ring> $K^0(X)\otimes\mathbb Z[1/n]$, the finite <geometric series>
$$
(n+\eta)^{-1}=\frac1n\sum_{j=0}^{N-1}\left(-\frac{\eta}{n}\right)^j
$$
is an inverse. The <projection formula for the K-theory transfer> gives
$$
p_!p^*(a)=[P]a=(n+\eta)a.
$$
Thus $p^*(a)=0$ implies $a=0$ after inverting $n$. More explicitly, $(n+\eta)^{-1}p_!$ is a left inverse, proving
$$
\boxed{p^*:K^0(X)\otimes\mathbb Z[1/n]\hookrightarrow K^0(Y)\otimes\mathbb Z[1/n].}
$$
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