Let $G$ be a topological group. Let $p:E\to B$ be a fibre bundle with fibre $F$ and let $G$ act continuously on $F$.
The way one would typically define whether $E$ has $G$ as its structure group is by introducing the notion of a $G$-atlas, i. e. a family of local trivialisations covering $B$ such that the transition functions are given by continuous maps $U_i\cap U_j\to G$.
Now, one defines an equivalence relation on the class of $G$-atlasses by declaring two atlasses to be equivalent if their union is a $G$-atlas. Then, $E$ is called a $G$-bundle if it is equipped with one of the equivalence classes.
My problem is the following:
In this definition, the transition functions $U_i\cap U_j\to G$ do not necessarily depend solely on the atlas. When proving that the equivalence relation is transitive, one needs a way to convert transition functions $U_i \cap U'_j \to G$ and $U'_j \cap U''_k \to G$ into transition functions $U_i\cap U''_k\to G$. This can be done through pointwise group multiplication. However, this depends on the (local) choice of $j$.
Since this (locally defined) product is in fact the correct transition function, it follows that, in the case of a faithful action, the choice of $j$ is irrelevant.
One could then define a $G$-bundle as a $G/N$-bundle where $N = \ker (G\to\mathrm{Homeo}(F))$. However, the problem here is that a transition function $U_i\cap U_j\to G/N$ can, a priori, not necessarily be lifted to one with codomain $G$, meaning that some bundles might be $G$-bundles when they shouldn't be.
What is the "correct" definition of a $G$-bundle if $G$ acts non-faithfully on the fibre?