Algebraic fixed point theorem
Solution 1:
The only fixed point theorem involving finite groups I know is the following:
$p$-group fixed point theorem: Let $G$ be a finite $p$-group acting on a finite set $X$. Then $|X^G| \cong |X| \bmod p$. In particular, if $|X| \not \equiv 0 \bmod p$, then $G$ has a fixed point.
For example, applied to the conjugacy action of a finite $p$-group on itself, we conclude that such a group has nontrivial center. We can get a statement of your form by asking that $G$ is a $p$-group and $f$ has order a power of $p$.
Other applications are given here.