The Modular Isomorphism Problem -- the alternative perspective on counterexamples

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 FG and FH over the field F=F₂ 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.