Let $X$ be a compact complex surface and let $D$ be a double cover of $X$. Let $\pi:D\to X$ be the double cover map (a 2:1) map. If $E$ is a vector bundle (rank at least 2) on $X$ with $c_1(E) = A$ and $c_2(E) = k$ (I want to think of this Chern class as an integer by integration), then what can I say about the Chern classes of $\pi^*(E)$?
What if the double cover is branched over a divisor? That is, the map $\pi$ is 2:1 everywhere except on some lines on the surface? I'm more interested in the second Chern class and suspect the answer is $c_2(\pi^*(E)) = 2k$ because I believe you can think of the number $k$ as representing $k$ points on the surface $X$, so these $k$ points will pull back to $2k$ points on $D$. But I'm not sure about this/don't fully understand the argument.