  In this article, we characterize the (covariant) isotropy groups of free,
finitely generated racks and quandles. As a consequence, we show that the usual
inner automorphisms of such racks and quandles are precisely those
automorphisms that are "coherently extendible". We then use this result to
compute the global isotropy groups of the categories of racks and quandles,
i.e. the automorphism groups of the identity functors of these categories.
