Principal bundle structures among second order frame bundles

Using a model for the bundle $\hat{\mathcal F}^2M$ of semi-holonomic second order frames of a manifold $M$ as an extension of the bundle ${\mathcal F}^2M$ of holonomic second order frames of $M$, we introduce in $\hat{\mathcal F}^2M$ a principal bundle structure over ${\mathcal F}^2M$, the structure group being the additive group $A_2(n)$ of skew-symmetric bilinear maps from ${\Bbb R}^2 \times {\Bbb R}^n$ into ${\Bbb R}^n$. The composition of the projection of that structure with the existing projection of the bundle $\tilde{\mathcal F}^2M$ of non-holonomic second order frames of $M$ over $\hat{\mathcal F}^2M$ provides a principal bundle structure in $\tilde{\mathcal F}^2M$ over ${\mathcal F}^2M$. These results close an existing gap in the theory of second order frame bundles.