The core group is an invariant of unoriented virtual links. We introduce a peripheral structure for the core group, in which the longitudes are sensitive to orientations. We show that the combination of the core group and its peripheral structure is equivalent, as a link invariant, to the combination of the $π$-orbifold group and its peripheral structure. Examples show that the peripheral structure of the core group can be used to verify noninvertibility of some knots and links.