On finite groups with bounded conjugacy classes of commutators

In 1954 B. H. Neumann discovered that if $G$ is a group in which all conjugacy classes have finite cardinality at most $m$, then the derived group $G^{\mathrm{\prime }}$ is finite of $m$-bounded order. In 2018 G. Dierings and P. Shumyatsky showed that if $\mathrm{\mid }{x}^G\mathrm{\mid }\le m$ for any commutator $x\in G$, then the second derived group $G^{\mathrm{\prime }\mathrm{\prime }}$ is finite and has $m$-bounded order. This paper deals with finite groups in which $\mathrm{\mid }{x}^G\mathrm{\mid }\le m$ whenever $x\in G$ is a commutator of prime power order. The main result is that $G^{\mathrm{\prime }\mathrm{\prime }}$ has $m$-bounded order.