On quasifree profinite groups

Recently, it has been shown by Harbater and Stevenson that a profinite group $G$ is free profinite of infinite rank $m$ if and only if $G$ is projective and $m$-quasifree. The latter condition requires the existence of $m$ distinct solutions to certain embedding problems for $G$. In this paper we provide several new non-trivial examples of $m$-quasifree groups, projective and non-projective. Our main result is that open subgroups of $m$-quasifree groups are $m$-quasifree.