Let $L/K$ be a finite, cyclic extension of number fields, say with $\mathrm{Gal}(L/K)=G$. In my context $G$ is actually of order $p$, an odd prime number, but let me state my question for every cyclic $G$.

Hilbert theorem 94 says that *if $L/K$ is everywhere unramified* (hence contained in the Hilbert class field $H$ of $K$), then the ''capitulation kernel'', namely the kernel of the natural map $\iota:Cl_K\to Cl_L$, has order divisibile by $\vert G\vert$. Some (but, to my knowledge, not many) generalizations have been proven, mainly removing the cyclicity assumption although losing something.

My question goes into another direction: if $L/K$ is allowed to ramify and is contained in the ray class field $H(\mathfrak{f})$ modulo some conductor $\mathfrak{f}$, what can we say about the capitulation kernel $\iota:Cl_K(\mathfrak{f})\to Cl_L(\mathfrak{F})$, where these groups are now the ray class groups modulo the respective conductors (and $\mathfrak{F}=\mathfrak{f}\mathcal{O}_L)$)? Can it be trivial?

I must confess my ignorance with respect to an even more basic question, namely: what is the state of the art concerning the Principal Ideal Theorem for ray class fields? Is it known – or false, or trivial, or... – that all classes in $Cl_K(\mathfrak{f})$ become principal (perhaps, only $\mathfrak{f}$-principal?) in $H(\mathfrak{f})$?