Let $E$ be a 1-dimensional integral scheme (if you need to assume more, do so). Let $\hat{E}$ be $Spec$ of the completion of a stalk of $E$ at some closed point (think for example $E=Spec(\mathbb{C}[t])$ and $\hat{E}=Spec(\mathbb{C}[[t]])$).

Say we are given a branched Galois cover of, say, $\mathbb{P}^1_{\hat{E}}$ with branch divisor $\hat{D}$ (the divisor will not be a prime divisor - meaning it's made up of several irreducibles). Say we are also given a divisor of $E$, $D$, such that $D \times_E \hat{E}=\hat{D}$. It's not obvious (and probably not true) that this cover extends to a cover of $\mathbb{P}^1_E$ with branch divisor $D$ (by extends I mean that I require the new cover to be Galois as well with the same group).

Is there a way to measure the obstruction? Can I calculate something to check if I may really extend the cover?

Assume whatever you need to in order to avoid wild ramification since I'm not interested in that. And if you can solve a particular case of this, assume whatever you want.