  The paper establishes new relationship between cohomology, extensions and
automorphisms of quandles. We derive a four term exact sequence relating
quandle 1-cocycles, second quandle cohomology and certain group of
automorphisms of an abelian extension of quandles. A non-abelian counterpart of
this sequence involving dynamical cohomology classes is also established, and
some applications to lifting of quandle automorphisms are given. Viewing the
construction of the conjugation, the core and the generalised Alexander quandle
of a group as an adjoint functor of some appropriate functor from the category
of quandles to the category of groups, we prove that these functors map
extensions of groups to extensions of quandles. Finally, we construct some
natural group homomorphisms from the second cohomology of a group to the second
cohomology of its core and conjugation quandles.
