Isomorphism in expanding families of indistinguishable groups

For every odd prime $p$ and every integer $n\geq 12$ there is a Heisenberg group of order $p^{5n/4+O(1)}$ that has $p^{n^2/24+O(n)}$ pairwise nonisomorphic quotients of order $p^{n}$. Yet, these quotients are virtually indistinguishable. They have isomorphic character tables, every conjugacy class of a non-central element has the same size, and every element has order at most $p$. They are also directly and centrally indecomposable and of the same indecomposability type. The recognized portions of their automorphism groups are isomorphic, represented isomorphically on their abelianizations, and of small index in their full automorphism groups. Nevertheless, there is a polynomial-time algorithm to test for isomorphisms between these groups.