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Assume a finite group $P$ of prime power order acts smoothly on a disk $D$. We know from the Smith theory that the fixed point set $D^P$ is$\mod p$ - acyclic, where $|P|=p^k$ for some $k\in\mathbb{Z}$.

I wonder, which additional assumptions on the action of $P$ on $D$ guarntee that $D^P$ is simply-connected. Is it true in general? If not, maybe under some additional conditions for the action?

I know that, a piori, the conditions we assume do not guerantee even that the first homology group of $D^p$ (which is the abelianization of the fundamental group of $D^P$) vanishes - it follows from the Universal Coefficient Theorem - $H_1(D^P)$ may be potentially isomorphic to $\mathbb{Z}_q$ with $\gcd(p,q)=1$ for example.

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