A topological covering $ p : \tilde{X} \to X$ is normal when the group of deck transformations acts transitively on the fibers of $ p$ . This is equivalent to the fact that $ p_* (\pi (\tilde{X}, \tilde{x}))$ is a normal subgroup of $ \pi (X, p (\tilde{x}))$ . Such coverings are also known as Galois or regular.

The universal covering is known to be always normal. There are many nonnormal coverings, but all the classical examples that I know give me the impression of being cooked up for the purpose and being more created laboratory than observed in the wild.

The question is whether some nonnormal coverings are known to arise naturally in some mathematical problems or theories. This means that the covering should be motivated by some problem or theory and that unfortunately it turns out not to be normal.