The Modular Isomorphism Problem -- the alternative perspective on counterexamples

As a result of impressive research arXiv:2106.07231, D. Garc\'ıa-Lucas, \'{A}. del R\'{i}o 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.