Near abelian profinite groups

A compact $p$-group $G$ ($p$ prime) is called near abelian if it contains an abelian normal subgroup $A$ such that $G/A$ has a dense cyclic subgroup and that every closed subgroup of $A$ is normal in $G$. We relate near abelian groups to a class of compact groups, which are rich in permuting subgroups. A compact group is called quasihamiltonian (or modular) if every pair of compact subgroups commutes setwise. We show that for $p \neq 2$ a compact $p$-group $G$ is near abelian if and only if it is quasihamiltonian. The case $p = 2$ is discussed separately.