Let $G$ act faithfully such that $|\mbox{fix}(g)| \in \{0,n\}$ for each $g \ne 1$. Then $|\Omega| \equiv n \pmod{|G_{\alpha}|}$.
This should be a corollary from two lemmata I will give; but I even do not get its meaning. As $G$ is not required to act transitive, the point stabilizers could have many different sizes. So I do not see how it is implied?
The other lemmata are:
Lemma 1: Let $G$ act faithfully and transitive on $\Omega$ such that $|\mbox{fix}(g)| \in \{0,n\}$ for $g \ne 1$. Then there are $$ \frac{|G|(|G_{\alpha}| - 1)}{|G_{\alpha}| \cdot n} $$ non-trivial elements of $G$ with fixed points in $\Omega$.
Proof: Counting pairs $\{ (\alpha, h) \in \Omega\times G : \alpha^h = \alpha \}$ in two ways, and using that the point stabilizers all have the same size, we have $$ |\Omega||G_{\alpha}| = kn + 1\cdot |\Omega| $$ where $k$ denotes the number of non-trivial element of $G$ with fixed points. $\square$
Lemma 2: Let $G$ act faithfully on $\Omega$ such that $|\mbox{fix}(g)| = n$ for $g \ne 1$. Then $|\Omega| \equiv n \pmod{|G|}$.
Proof: Again counting pairs $(\alpha, h)$ with $\alpha \in \Omega$ and $h \in G_{\alpha}$, we have, as each non-trivial element fixes $n$ points, $$ |\Omega| + n(|G| - 1) = \sum_{\alpha \textrm{ represents one orbits}} |G : G_{\alpha}||G_{\alpha}| = k|G| $$ where $k$ denotes the number of orbits. $\square$
Now the firstly stated property should somehow follow from these two lemmata.