I'm basically wondering how to make "curved" the first column of the diagram $\require{AMScd}$ \begin{CD} P_1 @>\textrm{inclusion} >> G\\ @V \omega_0 V P_1\cap P_2 V @V\omega V P_2 V\\ P_1/(P_1\cap P_2) @= G/P_2 \end{CD} where $M:=G/P_2$ is a compact homogeneous space, such that the smaller group $P_1\subset G$ still acts transitively on $M$. Above, I regard $M$ as a flat Cartan geometry of type $(G,P_2)$, with Maurer-Cartan form $\omega$.

Simultaneously, I want to regard $M$ as a (non necessarily flat) Cartan geometry of type $(P_1,P_1\cap P_2)$, which is **compatible** with the flat $(G,P_2)$-type one, in the sense that the first column of
$\require{AMScd}$
\begin{CD}
\mathcal{G}_\sigma @>\textrm{inclusion} >> G\\
@V \omega_\sigma V P_1\cap P_2 V @V\omega V P_2 V\\
M @= M
\end{CD}
is no longer flat, i.e., $\mathcal{G}_\sigma$ is a principal sub-bundle of $G$ (and not necessarily a subgroup).

Then I see a nonempty family (it contains at least the flat case) $$ \{(\mathcal{G}_\sigma\to M, \omega_\sigma)\}_{\sigma\in \boldsymbol{\Sigma}} $$ of compatible Cartan geometries of type $(P_1,P_1\cap P_2)$.

MAIN QUESTION:what is the space $\boldsymbol{\Sigma}$ parametrising these compatible "sub-Cartan geometries"?

*A rather brutal reasoning led me to believe that
$$
\boldsymbol{\Sigma}=\Gamma(\pi),
$$
where $\pi:G/N_{P_2}(P_1\cap P_2)\longrightarrow G/P_2=M$. Indeed, the generic fibre of $\pi$ is $P_2/N_{P_2}(P_1\cap P_2)$, which tells how many subgroups exist in $P_2$, isomorphic to $P_1\cap P_2$: hence, a section $\sigma$ of $\pi$ allows me to construct a principal $(P_1\cap P_2)$-subbundle $\mathcal{G}_\sigma$ of $G$.*

Finally, suppose that $\boldsymbol{\Sigma}$ admits a geometric description (similar, at least in spirit, to my brute attempt above) in terms of natural structures associated to $M$.

SIDE QUESTION:what is the correct way to "pull-back" the Maurer-Cartan form $\omega$ from $G$ to $\mathcal{G}_\sigma$ thus obtaining $\omega_\sigma$? (In the sense that it should be a pull-back which introduce somehow the curvature.)

If my question is well-posed, then I cannot believe that this matter has not been given attention before: any reference will be appreciated!