A fundamental theme in the study of group algebras is that what kind of properties of underlying groups can be detected from the structure of the algebras. The modular isomorphism problem, which was open for more than six decades, asks whether an isomorphism of group algebras of finite $p$-groups over the field with $p$ elements implies an isomorphism of their underlying groups. In this talk, we show that how unit groups of algebras and counting homomorphisms can be used to extract information about groups.