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)

As Franz Lemmermeyer suggested, one should consider the so-called Ambigous Class Number Formula: you find it, for instance, in Gras' book "Class Field Theory", II.6.2.3.

It says that if $L/K$ is a finite cyclic Galois extension with group $G$ and if we denote by $Cl_L,E_L$ the class group and the unit group of $L$ respectively (and likewise for $K$), then $$ \vert Cl_L^G\vert=\frac{\vert Cl_K\vert\cdot Ram(L/K)}{[L:K]\cdot [E_K:E_K\cap\mathrm{Norm}_{L/K}(L^\times)]} $$ where $Ram(L/K)$ is the product of all ramification indexes of finite and infinite primes of $K$ in the extension $L/K$.

So, in case $K=\mathbb{Q}$, you get simply $$ \vert Cl_L^G\vert=\frac{Ram(L/\mathbb{Q})}{[L:\mathbb{Q}]}\quad\text{or}\quad \vert Cl_L^G\vert=\frac{Ram(L/\mathbb{Q})}{2[L:\mathbb{Q}]} $$ according as whether $L^\times$ contains or not an element of norm $-1$.

The proof is basically cohomological following the one suggested by Joe Silverman: the term $Ram(L/K)$ comes from fixed ideals in $I_L^G$ and the index of the norm of units of $L$ inside those of $K$ controls some capitulation kernel - but you can look in Gras' book for details.

In case of dihedral extensions, you can look at the question Class groups in dihedral extensions - some sort of Spiegelungssatz?