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Suppose that $G$ is a finite $p$-group. If all subgroups of index $p^t$ of$G$ are abelian and at least one subgroup of index $p^{t-1}$ of $G$ is notabelian, then $G$ is called an $\mathcal{A}_t$-group. In this paper, someinformation about $\mathcal{A}_t$-groups are obtained and$\mathcal{A}_3$-groups are completely classified. This solves an {\it oldproblem} proposed by Berkovich and Janko in their book. Abundant informationabout $\mathcal{A}_3$-groups are given.

Finite $p$-groups all of whose subgroups of index $p^3$ are abelian