  We study the Modular Isomorphism Problem applying a combination of existing
and new techniques. We make use of the small group algebra to give a positive
answer for two classes of groups of nilpotency class 3. We also introduce a new
approach to derive properties of the lower central series of a finite $p$-group
from the structure of the associated modular group algebra. Finally, we study
the class of so-called $p$-obelisks which are highlighted by recent
computer-aided investigations of the problem.
