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.