Suppose $X / K$ is a variety over a finitely generated field over $\mathbb{Q}$. Fix an embedding $K \subset \mathbb{C}$ and let $\pi := \pi_1(X(\mathbb{C}), x)$ be the topological fundamental group. Spreading out to $A \subset K$ we can get specialization maps
$$ \pi \to \hat{\pi} = \pi_1^{\mathrm{et}}(X_{\mathbb{C}}, \bar{x}) \to \pi_1^t(X_{\bar{s}}, \bar{x}) \subset \pi_1^t(X_{s}, \bar{x}) $$
where $s \in \mathrm{Spec}(A)$ is a point of characteristic $p$. Suppose for $\ell \neq p$ I have an $\ell$-adic arithmetic representation:
$$ \rho : \pi_1^t(X_{\bar{s}}, \bar{x}) \to \mathrm{SL}_r(\overline{\mathbb{Q}}_\ell) $$
which for simplicity I have assumed is actually a representation of the full arithmetic fundamental group of $X_s$. Then for any $\ell' \neq \ell, p$ and a field isomorphism $\sigma : \overline{\mathbb{Q}}_{\ell} \to \overline{\mathbb{Q}}_{\ell'}$ there is a $\sigma$-companion: a continuous arithmetic $\ell'$-adic representation
$$ \rho^{\sigma} : \pi_1^t(X_s, \bar{s}) \to \mathrm{SL}_r(\overline{\mathbb{Q}}_{\ell'}) $$
compatible in the sense that the Frobenius characteristic polynomials are transported by $\sigma$.
My question is: if we compose with the specialization to get $\rho|_{\pi}$ and $\rho^{\sigma}|_{\pi}$ is there anything we can say about how these representations are related. Because $\pi$ is dense (in the geometric $\pi_1$), it is clear that $\rho|_{\pi}$ uniquely determines $\rho^{\sigma}|_{\pi}$ (up to conjugation since the extension to the arithmetic group is unique up to conjugation) but this is the extent of my knowledge.
A naive question is if the characteristic polynomials of $\rho|_{\pi}$ and $\rho^{\sigma}|_{\pi}$ are also transported by $\sigma$? This seems unlikely but I don't see a counterexample.
