Finite groups in which every commutator has prime power order (Q6597520)

Let \(G\) be a finite group, if every element of \(G\) has prime power order, then \(G\) is called an \(\mathsf{EPPO}\)-group (such a group is the set-theoretic union of its Sylow subgroups). If \(G\) is a soluble EPPO-group, then the Fitting height of \(G\) is at most 3 and \(|\pi (G)| \leq 2\) [\textit{G. Higman}, J. Lond. Math. Soc. 32, 335--342 (1957; Zbl 0079.03204)], see also [\textit{G. Zacher}, Rend. Sem. Mat. Univ. Padova 27, 267--275 (1957; Zbl 0166.28801)]. Moreover, \textit{M. Suzuki}, in [Trans. Am. Math. Soc. 99, 425--470 (1961; Zbl 0101.01604); Ann. Math. (2) 75, 105--145 (1962; Zbl 0105.25501)], showed that if \(G\) is insoluble, then the soluble radical \(R(G)\) of \(G\) is a 2-group and there are exactly eight nonabelian simple \(\mathsf{EPPO}\)-groups: \(\mathrm{PSL}(2,q)\) (\(q=4, 7, 8, 9, 17\)), \(\mathrm{PSL}(3,4)\), \(\mathrm{Sz(8)}\), and \(\mathrm{Sz}(32)\).\N\NIn the paper under review the authors focus on finite groups in which every commutator has prime power order (\(\mathsf{CPPO}\)-groups). They first prove that if \(G\) is a soluble \(\mathsf{CPPO}\)-group, then the Fitting height of \(G\) is at most 3 and the order of \(G'\) is divisible by at most 3 primes. Then they prove that, if \(G\) is an insoluble \(\mathsf{CPPO}\)-group, then (a) \(G'\) is perfect, (b) \([R(G)=[G,R(G)]\) is a 2-group and (c) \(G'/R(G')\) is a simple \(\mathrm{EPPO}\)-group.\N\NIn the paper the following question is proposed: is it true that if \(G\) is a \(\mathsf{CPPO}\)-group, then \(G'\) is an \(\mathsf{EPPO}\)-group?