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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.

A. Erschler, Not residually finite groups of intermediate growth, commensurability and non-geometricity, J. Alg. 272 (2004), 154--172.

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.

I have just deleted a clearly nonsensical answer!

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.

A. Erschler, Not residually finite groups of intermediate growth, commensurability and non-geometricity, J. Alg. 272 (2004), 154--172.

Suppose that $G$ is any countable group. Then there exists a short exact sequence $1 \to K \to G \to F \to 1$, where $F$ and $K$ are countable free groups. Clearly $F$ and $G$ have faithful linear representations over $\mathbb{C}$. However, there are many examples of countable groups with no faithful linear representations; e.g. infinite finitely generated simple groups.

Whoops! This is nonsense ... I need to drink my morning coffee.

I have just deleted a clearly nonsensical answer!