Relativization of symmetries on quandles

This paper introduces relative versions of the inner automorphism group and the transvection group associated with surjective quandle homomorphisms.By using the relative inner automorphism group, we define a notion of \emph{connectedness} for surjective homomorphisms. We characterize connected homomorphisms algebraically as quotient maps, and use the relative transvection group to establish a maximal \emph{connected-covering} factorization for arbitrary surjections. Finally, we study surjective homomorphisms for which the relative inner automorphism group acts $2$-transitively on each fiber. Under this assumption, we classify the possible quandle structures of the finite fibers.