Normal covers correspond to normal subgroups, and the deck group is the normalizer quotient $N_G(H)/H$.Suggest a slogan
Let \(p\colon(\tilde X, \tilde x_0)\to(X, x_0)\) be a covering space with \(\tilde X\) path-connected and \(X\) path-connected and locally path-connected. Let \(G\coloneqq\pi_1(X, x_0)\) and \(H\coloneqq p_*\pi_1(\tilde X, \tilde x_0)\), and let \(G(\tilde X)\) be the group of deck transformations. Then
\(\tilde X\) is normal if and only if \(H\normal G\);
\(G(\tilde X) \cong N_G(H)/H\), where \(N_G(H)\) is the normalizer of \(H\) in \(G\).
In particular, \(G(\tilde X) \cong G/H\) if \(\tilde X\) is normal, and \(G(\hat X) \cong G\) for the universal cover \(\hat X\) [@Hat02].