The Jordan property for finite subgroups of ${\rm GL}_n(\mathbb{C})$ says that there exists a constant $c(n)$ so that for any finite subgroup $G$ of ${\rm GL}_n(\mathbb{C})$ there is a normal abelian subgroup $A$ of $G$ of index at most $c(n)$, and we can embed $A$ in a maximal torus of ${\rm GL}_n(\mathbb{C})$.
I was wondering if there exists some sort of "quotient version". I am expecting some result of the form: There exists a constant $c(n)$ satisfying the following: Let $G$ be a subgroup of ${\rm GL}_N(\mathbb{C})$ and $T_1$ be a subtorus of rank $N-n$ of a maximal torus $\mathbb{G}_m^N$ of ${\rm GL}_N(\mathbb{C})$. Assume that there exists an exact sequence $1\rightarrow T_1\rightarrow G\rightarrow G/T_1\rightarrow 1$, where $G/T_1$ is a finite group. Then, there exists a normal abelian subgroup $A\leqslant G/T_1$ of index at most $c(n)$ so that its pre-image on $G$ lies in the maximal torus (or some conjugate).
If $T_1$ is the trivial torus, this gives back the Jordan property. I am not an expert on representation theory. Some results in algebraic geometry I am working on point out to a statement like this, so I wanted to know if there is anything in the literature.