Let $\widetilde {X}$ be the universal cover of a 'nice' space $X$. Suppose $H$ be a subgroup of Deck $(\widetilde {X},X)$.Is it true that Deck$(H \backslash \widetilde {X},X) \cong \frac {N(H)}{H}$ where $N(H)$ denote the normalizer of $H$ in Deck $(\widetilde {X},X)$?
I think its not true but unable to comp up with example.Any ideas?