Let $V$ be a vector space and $\textrm{GL}(V)$ the group of automorphisms on $V$.
Let $Z(V) \vartriangleleft \textrm{GL}(V)$ the normal subgroup of homotheties on $V$ (i.e. the centre of $\textrm{GL}(V)$).
Let $\pi: \textrm{GL}(V) \to \textrm{PGL}(V):=\textrm{GL}(V)/Z(V)$ be the canonical projection.
Let $H<\textrm{PGL}(V)$ be a finite subgroup.
I want to show that if $H$ is soluble(/solvable) then $\pi^{-1}(H)$ is soluble.
My attempt at a proof:
There exists a sequence of groups:
$$ \{\textrm{Id}\}=:H_0 \vartriangleleft H_1 \vartriangleleft ... \vartriangleleft H_{r-1} \vartriangleleft H_{r}:= H $$ such that $H_{i}/H_{i-1}$ is Abelian for all $i \in \{ 1,...,r \}$.
If we then let $G_i:= \pi^{-1}(H_i)$ for all $i \in \{0,...,r \}$, then we have a sequence
$$ \{\textrm{Id}\}=:G_0 \vartriangleleft G_1 \vartriangleleft ... \vartriangleleft G_{r-1} \vartriangleleft G_{r}:= G $$ as the holomorphic preimage of a normal subgroup is again normal.
My only difficulty lies in showing that $G_{i}/G_{i-1} = \pi^{-1}(H_i)/\pi^{-1}(H_{i-1})$ is Abelian for all $i \in \{ 1,...,r \}$.
Does anyone have any ideas as to how one would go about demonstrating this last bit / any counterexample?
This is related to my attempted proof of this statement, which, as was pointed out in a comment, is in fact false.
Thank you for your attention.