Let $\pi_P:P\rightarrow M$ a principal $G$ (right action) bundle. Let $F$ be a manifold with a left action of $G$. Then we have the notion of associated fibre bundle over $M$ whose fibre is $F$. I do not know if there is any connection in the other way which is an inverse for the above construction.

Suppose we start with a $Gl(n,\mathbb{R})$ bundle. We have associated vector bundle whose fibre is $\mathbb{R}^n$ which is a vector bundle.

Suppose we start with a vector bundle of rank $n$, we have the notion of what is called frame bundle which is a principal $Gl(n,\mathbb{R})$ bundle.

This says that there is a correspondence between vector bundle of rank $n$ and principal $Gl(n,\mathbb{R})$ bundle.

Do we have such correspondence in case of

fibre bundles? Given a fibre bundle, $\pi:E\rightarrow M$, with fibre $F$, can I produce a principal $G$ bundle with an action of $G$ on $F$ such thatassociated fibre bundleis precisely the one I started with?