  As a result of impressive research arXiv:2106.07231, D.~Garcia-Lucas, A. del
Rio and L. Margolis defined an infinite series of non-isomorphic $2$-groups $G$
and $H$, whose group algebras $\mathbb{F}G$ and $\mathbb{F}H$ over the field
$\mathbb{F}=\mathbb{F}_2$ are isomorphic, solving negatively the long-standing
Modular Isomorphism Problem (MIP). In this note we give a different perspective
on their examples and show that they are special cases of a more general
construction. We also show that this type of construction for $p>2$ does not
provide a similar counterexample to the MIP.
