The general linear group $\operatorname{GL}_{n+1}(\mathbb R)$ (as a Lie group) acts smoothly on the real projective space $\mathbb R \mathbb P^n$, via $A \cdot [x] := [Ax]$. By $[x]$ here, I mean the $1$-dimensional subspace of $\mathbb R^{n+1}$ spanned by $x \in \mathbb R^{n+1} \backslash\{0\}$. This action is transitive, so $\mathbb R \mathbb P^n$ is a homogeneous $\operatorname{GL}_{n+1}(\mathbb R)$-space.
A corollary of the Quotient Manifold Theorem is that if $M$ is a homogeneous $G$-space, then $G/G_p$ is $G$-equivariantly diffeomorphic to $M$, where $p$ is any point in $M$ and $G_p$ is the isotropy subgroup at $p$. In the case of $\operatorname{GL}_{n+1}(\mathbb R)$ acting on $\mathbb R \mathbb P^n$, is there a nice description of the isotropy subgroup at a point, say $[(0,\ldots,0,1)]$?