Let $p:E \to X$ a topological covering of connected space $X$. Fix a basepoint $x_0$ of $X$ and denote by $\pi(X,x_0) $ the fundamental group of $X$.
The monodromy action of $\pi(X,x_0) $ on $p^{-1}(x_0)$ is defined by lifting a loop representant from $\pi(X,x_0) $ starting in a choosen $y_0 \in p^{-1}(x_0)$.
If $E,X$ are nice enough the lifting theorem garantees the uniqueness of this lift.
I often read that some extra conditions (...like locally path-connectedness and so on) for $X$ garantee also the existence of a simply connected universal cover $X_U$ such that every cover $E$ of $X$ with nice enough properties obtains a map $f_E: X_U \to E$.
By construction of $X_U$ there exist a map $\pi(X,x_0) \to Aut(X_U \vert X)$ taking into account the choice of base point $x_0$.
Futhermore $f_E$ induces a map $Aut(X_U \vert X) \to Aut(E \vert X)$ and therefore we obtain a map $\pi(X,x_0) \to Aut(E \vert X)$.
My question is if there exist a way to visualize or understand intuitively/geometrically what this map does.
Indeed the action of $\pi(X,x_0)$ of the fiber $p^{-1}(x_0)$ has a very intuitive & geometric interpretation as explaned above. Therefore I have keen interest to find out how the automorphism $a_{\gamma} \in Aut(E \vert X)$ induced by a loop $ \gamma \in \pi(X,x_0) $ looks like. Can it be visualized and what intuition lies behind that?