Let $\Omega$ and $G$ and be finite and let $G$ act on $\Omega$. Define $\operatorname{Fix}_\Omega(g)=\{x \in \Omega : x^g=x\}$. I am trying to show that $\sum_{g\in G}|\operatorname{Fix}_\Omega(g)|=t|G|$, where $t$ is the number of orbits of $G$ on $\Omega$. I have shown that $\sum_{g\in G}| \operatorname{Fix}_\Omega(g)|=\sum_{\alpha \in \Omega}|G_\alpha|$. I am guessing that one would now apply the Orbit-Stabiliser theorem, but I am not sure how to continue from there.
EDIT: To make it clear, $G_\alpha$ is the stabiliser of $\alpha$ under the action (I made a notational error that I have now fixed).
Let $O_i$ be an orbit, and let $\alpha_1$ be a specific element of $O_i$. Then $\sum_{\alpha \in O_i}|G_\alpha| = |G:G_{\alpha_1}||G_{\alpha_1}|=|G|$. The reason is that $|O_i|=|G:G_{\alpha}|=|G:G_{\alpha_1}|$ since the cosets of $G_{\alpha}$ are in 1:1 correspondence with the elements of $O_i$. (And of course $|G_{\alpha}|=|G_{\alpha_1}|$ since they are conjugate by an $h$ that takes $\alpha$ to $\alpha_1$. Thus, each orbit contributes |G| to the sum $\sum_{\alpha \in \Omega}|G_\alpha|$.