Eigenspace of finite abelian group

Question

Let $\rho: G\to {\rm GL}_n(\mathbb{C})$ be faithfull representation of finite abelian group $G$ and $V$ is the eigenspace of some $g\in G$.

Is it true that $V$ is also eigenspace for all $G$ (that is $\rho(g)v=\lambda_g v$ for all $v\in V$ and $g\in G$)?

Answer 1

No.

For a counterexample, take any faithful representation of a nontrivial group (simplest is an action of $\Bbb Z_2$ on $\Bbb C^2$, say, by reflection through a line), and consider the unit element, that has the whole space as eigenspace, while other elements can have more complex eigenspace decomposition.