The fixed points of coprime action (Q1590712)

Let \(\pi\) be a set of odd primes, and suppose that the \(\pi\)-group \(P\) acts as a group of automorphisms of a solvable \(\pi'\)-group \(G\). Then, the author proves that the centralizer \(C_{[G,P]}(P)\) of \(P\) in the commutator group \([G,P]\) is generated by the union of the \(C_{[g,P]}(P)\) for \(g\in G\). The author points out that the conclusion of the theorem may be false if \(\pi\) is allowed to contain the prime \(2\).