Let $P(V)$ be a projective space and consider the natural action of $G=PGL(V)$ on it. Let $S=\{p_1,\dots, p_k\}$ be a finite set of points in $V$ where $k\geq 2$. Is there any reference about the subgroup $G_S$ of $PGL(V)$ stabilizing $S$ point-wisely?
Lift each $p_i$ to a nonzero vector $v_i\in V$. Let's first assume $v_i$ are all linearly independent. Then I think $G_S$ is the projective image of the subgroup of $GL(V)$ that preserves the directions of $v_i$. But $v_i$ could be scaled by different scalars.
On the other hand, if we add an extra vector $v_{k+1}=v_1+\dots+v_k$, and add the corresponding point $p_{k+1}$ to $S$, then $G_S$ becomes the projective image of the subgroup of $GL(V)$ that fixes the subspace spanned by $v_i$ point-wisely, since now all scalars must be the same: otherwise, suppose $v_i$ are scaled by $c_i$ and say $c_1\neq c_2$, then $v_{k+1}=\sum_{i=1}^k v_i$ is sent to $\sum_{i=1}^k c_i v_i$ which is not a multiple of $v_{k+1}$.
In general I think the situation is a mixture of the two above, and one need to decompose the set $S$ into the disjoint union of subsets in order to describe $G_S$. This corresponds to the decomposition of the matroid represented by $v_i$ into connected components.
My question is (1) is this correct? (2) is there a reference that works $G_S$ out? As the group $G_S$ is so natural, I think there are probably some standard references that cover it, but I can't find any.