Let $K$ be a number field and let $G$ be the group of automorphisms of $K$ over $\mathbf Q$. The group $G$ acts in a natural way on the ideal class group of $K$. I would like to know if there are any results giving a formula for the number of orbits of this action (or equivalently a formula for the number of ideal classes that are fixed by some element of $G$). In particular, I would like to compare the number of orbits to the class number of $K$.



Ideal classes fixed by the Galois group
I assume you want $K$ to be Galois over $\mathbb{Q}$. More generally, let $L/K$ be a Galois extension of number fields. The the class group $C\_K$ of $K$ maps to $C\_L^{G\_{L/K}}$, the part of $C\_L$ fixed by the Galois group of $L/K$, and you seem to be asking what the quotient $C\_L^{G\_{L/K}}/C\_K$ looks like.


Taking cohomology of the exact sequences
$$ 1\to R\_L^\*\to L^\*\to L^\*/R\_L\*\to1
\quad\text{and}\quad
 1\to L^\*/R\_L\* \to I\_L \to C\_L \to 1
$$
gives (if I'm not mistaken) exact sequences
$$
 0 \to H^1(G\_{L/K},L^\*/R\_L\*) \to H^2(G\_{L/K},R\_L^\*) \to \text{Br}(L/K)
$$
and
$$
 0 \to C\_K \to C\_L^{G\_{L/K}} \to H^1(G\_{L/K},L^\*/R\_L\*),
$$
so the quotient that you're interested in naturally injects
$$
 C\_L^{G\_{L/K}}/C\_K \hookrightarrow 
 \text{Ker}\Bigl(H^2(G\_{L/K},R\_L^\*) \to \text{Br}(L/K)\Bigr).
$$
The Galois structure of unit groups has been much studied. You might look at some of Ted Chinburg's papers (<http://www.math.upenn.edu/~ted/CVPubs9-10-07.html>)


