 Isometry group of a Riemannian quotient

For a complete simply connected Riemannian manifold $X$ and a free properly discontinuous deck group $\Gamma$, each isometry of $X/\Gamma$ lifts to an isometry of $X$ normalizing $\Gamma$. Conversely such a normalizing isometry descends; two lifts differ by a deck transformation. This proves the formula, including orientation-reversing isometries if the full normalizer is used.