  We study the Modular Isomorphism Problem (MIP) for groups of small order
based on an improvement of an algorithm described by B. Eick. Our improvement
allows to determine quotients $I(kG)/I(kG)^m$ of the augmentation ideal without
first computing the full augmentation ideal $I(kG)$. It allows us to verify
that the MIP has a positive answer for groups of order $3^7$ and to
significantly reduce the cases that need to be checked for groups of order
$5^6$. We further provide a proof for an observation of Bagi\'nski and provide
a negative answer to a question of Bleher, Kimmerle, Roggenkamp and Wursthorn.
