In Cossey and Stonehewer ['On the rarity of quasinormal subgroups', Rend. Semin. Mat. Univ. Padova 125 (2011), 81-105] it is shown that for any odd prime p and integer n >= 3, there is a finite p-group G of exponent p(n) containing a quasinormal subgroup H of exponent p(n-1) such that the nontrivial quasinormal subgroups of G lying in H can have exponent only p, p(n-1) or, when n >= 4, p(n-2). Thus large sections of these groups are devoid of quasinormal subgroups. The authors ask in that paper if there is a nontrivial subgroup-theoretic property X: of finite p-groups such that (i) X is invariant under subgroup lattice isomorphisms and (ii) every chain of X-subgroups of a finite p-group can be refined to a composition series of X-subgroups. Failing this, can such a chain always be refined to a series of X-subgroups in which the intervals between adjacent terms are restricted in some significant way? The present work embarks upon this quest
