Exact sequences in the cohomology of a Lie superalgebra extension

Let $ 0rightarrow mathfrak{a} rightarrow mathfrak{e} rightarrow mathfrak{g} rightarrow 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 is our main result, involving various endomorphism groups and the second cohomology. 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.