Consider $L/K$ with Galois group $G$. Define the symmetrisation of $H$ (subgroup of $G$) to be: $$ \phi_H(x) \;=\; \sum_{h\in H} h(x). $$
Obviously, this is fixed by H.Moreover, in the case that the characteristic of $L$ is zero,then $$im(\phi_H)=L^H.$$ This is all contained in Jim Belk's answer in this
It often happens that if $\alpha$ generates $L$ over $K$, then $\phi_H(\alpha)$ generates $L^H$ over $K$. In Gerry Myerson's answer here, he says that you have to be unlucky for this to not be the case. My question is this: exactly how unlucky do you have to be, i.e. in which cases does this not happen?