Is a covering space of a topological group a fiber bundle with the structure group the fundamental group of the topological space? 
Question. Is a covering space of a topological group a fiber bundle with the structure group the fundamental group of the topological space?

Let $p:E\rightarrow X$ be a covering space of X. I know how $\pi_1(X,x_0)$ acts on $p^{-1}(x_0)$. And if the space is path-wise connected all the $\pi_1(X,x_0)$ are all isomorphic for all $x_0\in X$. Thus $\pi_1(X)$ acts on the fibers.
Am I correct or am I missing something? 

Yes. In fact, any regular covering space $p\colon E\to B$ is a princpal bundle with structure group given by $\pi_1(B)/p_*(\pi_1(E))$. So if $E$ is a universal cover (which is always regular), then the structure group is just $\pi_1(B)$.
The action is given in the usual way via the monodromy action (unique lifting of paths).
