  Birget, Margolis, Meakin and Weil proved that a finitely generated subgroup
$K$ of a free group is pure if and only if the transition monoid $M(K)$ of its
Stallings automaton is aperiodic. In this paper, we establish further
connections between algebraic properties of $K$ and algebraic properties of
$M(K)$. We mainly focus on the cases where $M(K)$ belongs to the pseudovariety
$\overline{\boldsymbol{\mathbf{{H}}}}$ of finite monoids all of whose subgroups
lie in a given pseudovariety $\overline{\boldsymbol{\mathbf{{H}}}}$ of finite
groups. We also discuss normal, malnormal and cyclonormal subgroups of $F_A$
using the transition monoid of the corresponding Stallings automaton.
