First, let's fix some definitions.
Definition 1: Let $G$ be a topological group. A principal G-bundle is a triple $\xi = (E,\pi,B)$, where $E$ is a right $G$-space and the action of $G$ on $E$ is free and such that $B\cong E/G$ and $\pi:E \to B$ is the quotient map under this identification. We also require a local triviality condition: for each $b\in B$ there is an open set $V$ containing $b$ and a $G$-equivariant homeomorphism $\psi:\pi^{-1}(U)\to U\times G$ such that $\psi\circ\pi = p_U\circ\psi$, where $p_U:U\times G\to U$ is the canonical projection.
Definition 2: Let $F$ be a topological space. A fiber bundle with fiber $F$ is a map $p:Y \to X$ such that for each $x\in X$ there is an open set $U$ containing $x$ and a homeomorphism $\phi:p^{-1}(U) \to U\times F$ such that $\phi\circ p = p_U\circ \phi$, where $p_U:U\times F\to U$ is the canonical projection.
With these definitions in mind, I'm curious if, for a fixed base space $X$, there is a bijective correspondence between fiber bundles with fiber $F$ and principal Aut$(F)$-bundles, where Aut$(F)$ is the group of self-homeomorphisms of $F$, say under the compact-open topology.
We have constructions going in each direction.
Start with a fiber bundle with fiber $F$, $p:Y\to X$. This gives rise to a trivializing open cover $\{(U_i,\phi_i)\}$, which gives rise to transition functions $\phi_{ij} = \phi_i\circ\phi_j^{-1}:(U_i\cap U_j)\times F\to (U_i\times U_j)\times F$, which we can view, by an application of the universal property of products and the $\times$-Hom adjuction (for suitably nice spaces) as a continuous map $\phi_{ij}:U_i\cap U_j\to$ Aut$(F)$. We can then form a principal Aut$(F)$-bundle by the usual construction. Take as our total space $$E=\bigsqcup_i (U_i\times \textrm{Aut}(F))/\sim,$$ where $U_i\times \textrm{Aut}(F)\ni (x,g)\sim (x',g')\in U_j\times\textrm{Aut}(F)$ if and only if $x = x'$ and $g' = \phi_{ij}(x)g$. One can check that the projections $U_i\times\textrm{Aut}(F)\to U_i$ glue to a projection $ \pi:E\to X$ and that we indeed get a principal Aut$(F)$-bundle.
Starting with a principal Aut$(F)$-bundle $\xi = (E,\pi,X)$, we can construct a fiber bundle with fiber $F$ by taking the product space $E\times F$ and dividing out by the Aut$(F)$-action given by $(e,f)\cdot g = (e\cdot g, g^{-1}\cdot f)$. The resulting space, $Y$, is our total space, and the projection $\pi:E \to X$ descends to a projection. $p:Y\to X$.
The question is then, are these two construction inverses? It very well may be that this is written up somewhere, but I wasn't able to find anything. Any help is appreciated.