Let $0\to \mathfrak{a}\to \mathfrak{e}\to \mathfrak{g}\to 0$ be an abelian extension of the Lie superalgebra $\mathfrak{g}$. In this article we consider the problems of extending endomorphisms of $\mathfrak{a}$ and lifting endomorphisms of $\mathfrak{g}$ to certain endomorphisms of $\mathfrak{e}$. We connect these problems to the cohomology of $\mathfrak{g}$ with coefficients in $\mathfrak{a}$ through construction of two exact sequences which are our main results. The first exact sequence is obtained using the Hochschild–Serre spectral sequence corresponding to the above extension while to prove the second we rather take a direct approach. As an application of our results we obtain descriptions of certain automorphism groups of semidirect product Lie superalgebras.

让$0\to \mathfrak{一种}\to \mathfrak{电子}\to \mathfrak{G}\to 0$是李超代数的阿贝尔扩展$\mathfrak{G}$. 在本文中，我们考虑扩展自同态的问题$\mathfrak{一种}$和提升自同态$\mathfrak{G}$对某些自同态$\mathfrak{电子}$. 我们将这些问题与上同调联系起来$\mathfrak{G}$系数在$\mathfrak{一种}$通过构建两个精确序列，这是我们的主要结果。第一个精确序列是使用与上述扩展对应的 Hochschild–Serre 谱序列获得的，而为了证明第二个序列，我们宁愿采用直接方法。作为我们结果的应用，我们获得了半直积李超代数的某些自同构群的描述。