  The question whether non-isomorphic finite $p$-groups can have isomorphic
modular group algebras was recently answered in the negative by Garc\'ia-Lucas,
Margolis and del R\'io [J. Reine Angew. Math. 783 (2022), pp. 269-274]. We
embed these negative solutions in the class of two-generated finite $2$-groups
with dihedral central quotient, and solve the original question for all groups
within this class. As a result, we discover new negative solutions and simple
algebra isomorphisms. At the same time, the positive solutions for most of the
groups in this class give some insights what makes the negative solutions
special.
