Hiss–O'Brien conjecture on invariant transversals and commutators

Let $G$ be a finite group, and let $H\leq G$ be an abelian subgroup. A $G$-invariant transversal for $H\backslash G$ is a set of representatives for the right $H$-cosets that is invariant under conjugation by $G$. Write $G'$ for the commutator subgroup of $G$.

Hiss–O'Brien conjecture. If $H$ admits a $G$-invariant transversal, then

$$H\cap G'=\{1\}.$$

The conjecture concerns the existence of conjugation-invariant coset representatives for abelian normal subgroups. The paper's abstract states that the conjecture has counterexamples when $H$ is abelian and $G$ is finite, so the claim is refuted.