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In the given generality, I'm not sure the question has much hope for a tidy answer.

Consider $G=GL(V)$ and write $V = W \oplus W'$ where $W$ has dimension 2. So you get an embedding $H=GL(W) \to G$ in a natural way .($H$ acts trivially on $W'$).

The stabilizer $Q$ in $H$ of a line $L \subset W$ is a parabolic subgroup of $H$. And the parabolic subgroups of $G$ are the stablizers stabilizers of flags $F$ in $V$. Now, the stablizer $P$ in $G$ of any

Consider a flag $$F = (0 \subset F_1 \subset F_2 \subset \cdots \subset F_r = V)$$ for which (i) $F_1 = L$, (ii) $F_2 = W$, and (iii) $F_i \cap W'$ is a complement in $F_i$ to $F_2=W$ has for $i \ge 2$.

If $P$ is the stabilizer of $F$ in $G= GL(V)$, then $P \cap H = Q$. And (Note that $H$ is contained in the stabilizer of the flag $F'=(0\subset F_2 \subset \cdots \subset F_r = V)$).

In general there are many such $P$.

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In the given generality, I'm not sure the question has much hope for a tidy answer.

Consider $G=GL(V)$ and write $V = W \oplus W'$ where $W$ has dimension 2. So you get an embedding $H=GL(W) \to G$ in a natural way.

The stabilizer $Q$ in $H$ of a line $L \subset W$ is a parabolic subgroup of $H$. And the parabolic subgroups of $G$ are the stablizers of flags $F$ in $V$. Now, the stablizer $P$ in $G$ of any flag $$F = (0 \subset F_1 \subset F_2 \subset \cdots \subset F_r = V)$$ for which $F_1 = L$ and $F_2=W$ has $P \cap H = Q$. And in general there are many such $P$.

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In the given generality, I'm not sure the question has much hope for a tidy answer.

Consider $G=GL(V)$ and write $V = W \oplus W'$ where $W$ has dimension 2. So you get an embedding $H=GL(W) \to G$ in a natural way.

The stabilizer $Q$ in $H$ of a line $L \subset W$ is a parabolic subgroup of $H$. And the parabolic subgroups of $G$ are the stablizers of flags $F$ in $V$. Now, the stablizer $P$ in $G$ of any flag $$F = (0 \subset F_1 \subset F_2 \subset \cdots \subset F_r = V)$$ for which $F_1 = L$ has $P \cap H = Q$. And in general there are many such $P$.