Isomorphism of commutative group algebras of closed $p$-groups and $p$-local algebraically compact groups

Let $G$ be an abelian group and let $K$ be a field of $\mathrm{char} K=p>0$. It is shown via a universal algorithm that if the modified Direct-Factor Problem holds, then the $K$-isomorphism $KH\cong KG$ for some group $H$ yields $H\cong G$ provided $G$ is a closed $p$-group or a $p$-local algebraically compact group. In particular, this is the case when $G$ is closed $p$-primary of arbitrary power, or $G$ is $p$-local algebraically compact with cardinality at most $\aleph_1$ and $K$ is in cardinality not exceeding $\aleph_1$. The last claim completely settles a question raised by W. May in Proc. Amer. Math. Soc. (1979) and partially extends our results published in Rend. Sem. Mat. Univ. Padova (1999) and Southeast Asian Bull. Math. (2001).