A quandle is an algebraic system with a binary operation satisfying three axioms modelled on the three Reidemeister moves of planar diagrams of links in the 3-space. 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.

quandle 是一个具有二元运算的代数系统，它满足三个公理，该公理以 3 空间中链接的平面图的三个 Reidemeister 移动为模型。论文在quandles的上同调、外延和自同构之间建立了新的关系。我们推导出四项精确序列，这些序列与 quandle 1-cocycles、second quandle 上同调和 quandles 的阿贝尔扩展的自同构群有关。还建立了涉及动态上同调类的该序列的非阿贝尔对应物，并给出了提升四元自同构的一些应用。将群的共轭、核心和广义 Alexander quandle 的构造视为从 quandles 范畴到群范畴的某个适当函子的伴随函子，我们证明这些函子将群的扩展映射到四元组的扩展。最后，我们构造了一些自然群同态，从一个群的第二个上同调到它的核和共轭问题的第二个上同调。