Let \(p\colon \tilde X \to X\) be a covering space and \(x_0 \in X\). For \(\tilde x\in \inverseof{p}(x_0)\) and a loop \(\gamma\) in \(X\) based at \(x_0\), let \(\tilde\gamma\) be the unique lift of \(\gamma\) with \(\tilde\gamma(0) = \tilde x\). The monodromy action is the right action of \(\pi_1(X, x_0)\) on \(\inverseof{p}(x_0)\) given by \[ \tilde x \cdot [\gamma] \coloneqq \tilde\gamma(1) .\]
The endpoint \(\tilde\gamma(1)\) depends only on the homotopy class \([\gamma]\), by the homotopy lifting property of covering spaces. The stabilizer of \(\tilde x\) is \(p_*\pi_1(\tilde X, \tilde x)\). If \(X\) is path-connected, the action is transitive if and only if \(\tilde X\) is path-connected, and then \(\tilde x\cdot[\gamma]\mapsto p_*\pi_1(\tilde X, \tilde x)\,[\gamma]\) is a bijection from \(\inverseof{p}(x_0)\) onto the set of right cosets of \(p_*\pi_1(\tilde X, \tilde x)\) in \(\pi_1(X, x_0)\).
[@Hat02].