Let $X$ have sufficient adjectives, and $p_1,p_2:\tilde{X}_1,\tilde{X}_2 \to X$ be two covering spaces with the same monodromy action $\rho: \pi_1(X,x_0) \to S_n$. Then $p_1,p_2$ are isomorphic.
I am unsure if the above proposition is true. Here is my attempted proof:
Let $\tilde{x}_i \in p_i^{-1}(x)$, $i = 1,2$ be the elements which correspond to 1 when we think of the $\pi_1(X,x_0)$ as acting on the set $\{1,\ldots,n\}$ (ie the elements of the fiber labeled 1).
$\ast \ast$ The subgroup $\{\gamma \in \pi_1(X,x_0) \mid \rho(\gamma)(1) = 1\}$ is the image of $\pi_1(\tilde{X}_i,\tilde{x}_1)$ under $p_i$, as it is the set of all loops in $X$ which lift to loops at $\tilde{x}_i$.
Then, $(p_1)_\ast(\tilde{X}_1,\tilde{x}_1) = (p_2)_\ast(\tilde{X}_2,\tilde{x}_2)$, so the covers are isomorphic by the classification thoerem.
Is this proof correct? I am unsure about $\ast \ast$.