On finite groups with exactly one noncommutator

An element $x$ of a group $G$ is a commutator if it can be expressed in the form $x=a^{-1}{b}^{-1}ab$ for some $a,b\in G$. In 2010 MacHale posed the following problem in the Kourovka notebook: does there exist a finite group $G$, with $\mathrm{\mid }G\mathrm{\mid }>2$, such that there is exactly one element of $G$ which is not a commutator? We answer this question in the affirmative and provide an infinite series of such groups, the smallest group in our construction having size $16609443840$.