Let $F/L/K$ be field extensions with $L/K$ finite. Let $H=\text{Hom}_K(L,F)$ be the set of field homomorphisms $L\rightarrow F$ that fix $K$. Take $\alpha\in L$, and let $\lbrace \alpha_1,\dots,\alpha_n\rbrace=\lbrace \sigma(\alpha):\sigma\in H\rbrace$. Under what circumstances is $n_i=|\lbrace \sigma\in H:\sigma(\alpha)=\alpha_i\rbrace|$ constant for all $i$?
More generally, fix an intermediate field $L/M/K$, and let $\lbrace \tau_1,\dots,\tau_m\rbrace=\lbrace \sigma|_M:\sigma\in H\rbrace$. Under what circumstances is $m_i=|\lbrace \sigma\in H:\sigma|_M=\tau_i\rbrace|$ constant for all $i$?
For example, if $L/K$ is a normal extension, then $\sigma(L)\subseteq L$, and so $\sigma(L)=L$ since $L/K$ is algebraic. Then, $H=\text{Aut}_K(L)$ is a group, and it is easy to see that $n_i$ and $m_i$ are constant for all $i$. I would like to know how much this condition can be weakened.