  A finite group $G$ is called Involutive Yang-Baxter (IYB) if there exists a
bijective 1-cocycle $\chi: G \longrightarrow M$ for some $\mathbb Z G$-module
$M$. It is known that every IYB-group is solvable, but it is still an open
question whether the converse holds. A characterization of the IYB property by
the existence of an ideal $I$ in the augmentation ideal $\omega \mathbb Z G$
complementing the set $1-G$ lead to some speculation that there might be a
connection with the isomorphism problem for $\mathbb Z G$. In this paper we
show that if $N$ is a nilpotent group of class two and $H$ is an IYB-group of
order coprime to that of $N$, then $N\rtimes H$ is IYB. The class of groups
that can be obtained in that way (and hence are IYB) contains in particular
Hertweck's famous counterexample to the isomorphism conjecture as well as all
of its subgroups. We then investigate what an IYB structure on Hertweck's
counterexample looks like concretely.
