Suppose that $G$ is a Lie group and $H$ is a closed subgroup of $G$. Then it is well known that the projection $$\pi:G\to G/H$$ has the structure of a locally trivial fibre bundle with fibre $H$. As pointed out in the comments, it is usually understood to be a principal $H$-bundle.
However, in Bredon's book, Topology and Geometry, chapter $2$, section $13$, the author's first exercise is to show that $\pi$ is a fibre bundle with structure group $N(H)/H$, where $N(H)$ is the normalizer of $H$ in $G$?
The question is why the structure group here is $N(H)/H$?
In particular this seems to be in contradiction to some of the discussion in the comments. The following is a copy of the relevant page from Bredon's book which contains all related informations (before this exercise, there is a detailed discussion of the construction of local trivializations):
