**Let a finite cyclic group $G = \mathbb Z/n$ act continuously on an open $d$-ball $B^d$. Suppose further that this action extends to the closed ball $\overline{B^d}$. Is there necessarily a fixed point in the interior?**

Note that by Brouwer's fixed point theorem, there has to be some fixed point in the closed ball.

It is also true that for $n$ a prime power, every action of $\mathbb Z/n$ on an open ball has a fixed point, this can be seen from Smith theory.

On the other hand, for $n$ not a prime power, there are actions of $\mathbb Z/n$ on open balls without fixed points, see e.g. Theorem 8.3 in Bredon: Introduction to Compact Transformation Groups. But I am not aware of examples of such actions that extend to the closed ball.