$\DeclareMathOperator\Gl{Gl}$As I understand it a natural bundle is one for which a diffeomorphism on the base space lifts to an automorphism on the total space of the bundle. It is my understanding that given a smooth $n$-dimensional manifold $M$, every natural bundle over $M$ is an associated bundle to some higher order frame bundle $F^{k}M$.
If one iteratively takes the tangent frame bundle of $M$ k-times, these are natural bundles over $M$. I am trying to figure out how they can be associated to the higher order frame bundles, or more generally higher order nonholonomic frame bundles $\tilde{F^{k}}M$.
First order is fairly simple, we have that $FM=F^{1}M=\tilde{F^{1}}M$ and they are simply the same bundle.
Second order is more involved. We have that $FFM$ is a principal $\Gl(n+n^{2})$ bundle over $FM$ and a principal $\Gl(n+n^{2},\mathbb{R})\times \Gl(n,\mathbb{R})$ bundle over $M$.
The second order frame bundle $F^{2}M$ has a different structure group $G^{2}$; however in “Differential Geometry of Frame bundles” it's clearly shown how elements of $G^{2}$ can be described as embedded within elements of $\Gl(n+n^{2})$.
Given the natural way in which the iterative frame bundle is constructed, I'm sure there exists a simple way to relate them to the higher order (especially the nonholonomic) frame bundles. I say especiall nonholonomic jets because they are also constructed by an iterative process (i.e. prolongation). Can anyone help me here?
