TITLE: Are extensions of linear groups linear?
QUESTION [18 upvotes]: A group $G$ is said to be linear if there exists a field $k$, an integer $n$ and an injective homomorphism $\varphi: G \to \text{GL}_n(k).$ 
Given a short exact sequence 
$1 \to K \to G \to Q \to 1$ of groups where $K$ and $Q$ are linear (over the same field), is it true that $G$ is linear too?
Background: Arithmetic groups are by definition commensurable with a certain linear group, so they are finite extensions of a linear group, and finite groups clearly are linear (over any field).

REPLY [11 votes]: Erschler has shown that there exists a central extension $G$ of 
$\mathbb{Z} \mathbin{wr} \mathbb{Z}$ by a finite group $F$ which
is not residually finite. Thus the short exact sequence $1 \to F \to G \to \mathbb{Z} \mathbin{wr} \mathbb{Z} \to 1$
provides an example of a non-linear group which is an extension of two linear groups over $\mathbb{C}$.
A. Erschler, Not residually finite groups of intermediate growth,
commensurability and non-geometricity, J. Alg. 272
(2004), 154--172.