The Modular Isomorphism Problem – the alternative perspective on counterexamples

As a result of impressive research [5], D. García-Lucas, Á. del Río and L. Margolis defined an infinite series of non-isomorphic 2-groups G and H, whose group algebras $FG$ and $FH$ over the field $F=F_2$ 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.