Let $n$ be a positive integer, and $p$ a prime number. Let $K_i$ be the cyclotomic field containing exactly the $np^i$th roots of unity. Let $H$ be the inverse limit of $p$-power torsion of the class groups of the $K_i$. Let $V$ be the $\mathbb{Q}_p$ - vector space $H \otimes_{\mathbb{Z}_p} \mathbb{Q}_p$. It is a result of Iwasawa theory that $V$ is finite-dimensional.

For any integer $q$ that is relatively prime to $np$, we have an automorphism $f_q$ of $\bigcup_{i = 1}^{\infty} K_i$ which, for each root of unity $\omega$, brings $\omega^q$ to $\omega$. This induces an automorphism of $H$ and thus an automorphism of $V$, say $f^*_q$.

What I'm interested in is whether $\mathrm{Tr}(f^*_q) \in \mathbb{Q}_p$ is rational, and related questions. Are there situations where we know this is the case, besides obvious cases like $q = 1$ or $V$ is trivial etc.? Are there situations where we know this isn't the case? Are there related operators on $V$ that nontrivially have rational trace? Can we ever say something weaker like $\mathrm{Tr}(f_q)$ is algebraic over $\mathbb{Q}$?

The reason I ask: Given a smooth projective curve $C$ over a finite field $K$ with algebraic closure $\overline{K}$, the $q$th power Frobenius morphism on $C$ induces the $\overline{K}$-linear Frobenius morphism on $\overline{C} = C \otimes_{K} \overline{K}$, which induces an endomorphism of the $\ell$-adic cohomology vector space $H^1(\overline{C}, \mathbb{Q}_{\ell})$. The traces of powers of this morphism yield information about the zeta function of the curve, but the only reason this works is that these traces are rational numbers (more specifically, integers) so they can be seen as both $\ell$-adic numbers or as real numbers, the latter being what we need for applications to the zeta function.

To relate this to the original question, I don't think these $f_q$ morphisms are analogous to powers of the $\overline{K}$-linear Frobenius, but rather a closely related morphism, which could be briefly described as the "inverse Frobenius on coefficients", and whose powers still induce endomorphisms of the $\ell$-adic cohomology vector spaces which have rational traces which give us essentially the same information as the $\overline{K}$-linear Frobenius. So basically I want to know if a similar thing holds in the number field setting, with $V$ playing the role of $H^1(\overline{C}, \mathbb{Q}_{\ell})$, in hopes that it might yield interesting information pertaining to non-p-adic L-functions (although I don't expect anything nearly as straightforward as in the function field case).