The modular isomorphism problem and abelian direct factors

Let $p$ be a prime and let $G$ be a finite $p$-group. We show that the isomorphism type of the maximal abelian direct factor of $G$, as well as the isomorphism type of the group algebra over $\mathbb F_p$ of the non-abelian remaining direct factor, are determined by $\mathbb F_p G$, generalizing the main result in arXiv:2110.10025 over the prime field. In order to do this, we address the problem of finding characteristic subgroups of $G$ such that their relative augmentation ideals depend only on the $k$-algebra structure of $kG$, where $k$ is any field of characteristic $p$, and relate it to the modular isomorphism problem, reproving and extending some known results.