Suppose $F \subset E$ is a sub-bundle of E,where one uses "frames of $E$ adapted to $F$", meaning that the first $p$ components of the frame are a frame of $F$ (where we denoted by $r$ and $p$ the ranks of $E$ and $F$, respectively). Let $\imath : \operatorname{GL}_p \to\operatorname{GL}_r $ be the inclusion map, defined by:
$$ \operatorname{GL}_p \ni A \to \begin{pmatrix} A & 0\\ 0 & I_{(r-p)\times (r-p)} \end{pmatrix}$$
We know that $\operatorname{Fr}(F)$ is a $\operatorname{GL}_p $-principal bundle, I hope to construct a vector bundle isomorphism such that:
$$\operatorname{Fr}(E) \cong \imath_*(\operatorname{Fr}(F))$$ where $$\imath_*(\operatorname{Fr}(F)) = \{[(x,e_x),g]\quad | \quad g \in \operatorname{GL}_r, \{e_x\}-\text{frame of F}\}$$ and the left action of $\operatorname{GL}_p$ is defined by: $$A\cdot((x,e_x), g) = \left((x, (e_x\cdot A)), \begin{pmatrix} A & 0\\ 0 & I_{(r-p)\times (r-p)} \end{pmatrix} g\right)$$ Does the isomorphism between $Fr(E)$ and $\imath_*(\operatorname{Fr}(F))$ exist?