  Let $R$ be a finite unital commutative ring. We introduce a new class of
finite groups, which we call hereditary groups over $R$. Our main result states
that if $G$ is a hereditary group over $R$ then a unital algebra isomorphism
between group algebras $RG \cong RH$ implies a group isomorphism $G \cong H$
for every finite group $H$.
  As application, we study the modular isomorphism problem, which is the
isomorphism problem for finite $p$-groups over $R = \mathbb{F}_p$ where
$\mathbb{F}_p$ is the field of $p$ elements. We prove that a finite $p$-group
$G$ is a hereditary group over $\mathbb{F}_p$ provided $G$ is abelian, $G$ is
of class two and exponent $p$ or $G$ is of class two and exponent four. These
yield new proofs for the theorems by Deskins and Passi-Sehgal.
