  Let $ 0\rightarrow \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.
