Operation of a p'-group on a set of p-power order and fix points The question is related to Taft's Theorem about G-invariant radical complements. Let $A$ be an associative unitary finite-dimensional $K$-Algebra posessing a separable factor Algebra by ist nilradical. Suppose $G$ is a finite group such that the order of $G$ is not divisible by the characteristic of $K$ and $G$ is acting on $A$ as auto- or anti-automorphism. Taft has proven that a $G$-invariant radical complement exists. A $G$-invariant radical complement is a fix point of the set of all radical complements under the Action of $G$.
If we assume a finite field, then one can prove that the number of radical complements is a power of the characteristic of $K$. A natural question is:
Let $G$ be a finite $p'$-Group acting on a set $M$ of $p$-power order. Is there a fix Point in $M$ under $G$?
Within $S_3$ one can construct examples of a subgroup of order 2 acting on the cosets of $S_3$ Modulo another subgroup of order 2 -- which is a set containg 3 elements. Within this example no fix Point exist.
My questions are: 
1.) What is so Special within the mentioned situation that a fix point exists.
2.) Is there a General result on fix Points related to the described context?
 A: I think the guess in your comment is correct: There is the following result of Glauberman:

Theorem (Glauberman). Let the finite group $G$ act on the finite group $N$ by automorphisms, where $(\lvert G \rvert, \lvert N \rvert ) = 1$. Let both $G$ and $N$ on a set $\Omega$, where the action of $N$ is transitive, and the compability condition 
  $$ (\omega n) g = (\omega g) n^g \quad \text{for all } 
   \omega\in \Omega,\, n\in N,\, g\in G $$
  holds. Then $\Omega$ has a fixed point under $G$. Moreover, $C_N(G)$ acts transitively on the $G$-fixed points.  

In your situation, this lemma applies with $N = 1 + \operatorname{rad}(A)$, which acts transitively on the set of complements of the radical, if I understand correctly. 
Glauberman's result appears as Lemma 3.24 in Isaacs's text on finite groups or in Glauberman's original paper (Theorem 4).
Isaacs, I. Martin, Finite group theory. Graduate Studies in Mathematics 92. AMS 2008. ZBL1169.20001.
Glauberman, George, Fixed points in groups with operator groups, Math. Z. 84, 120-125 (1964). ZBL0123.02601.
