Let 0 → a → e → g → 0 be an abelian extension of Lie superalgebras. In this article, corresponding to this extension we construct two exact sequences connecting the various automorphism groups and the 0-th homogeneous part of the second cohomology group, H2(g, a)0 . These exact sequences constitute an analogue of the well-known Wells exact sequence for group extensions. From this it follows that the obstruction for a pair of automorphism (ℙ, ℙ) 2 Aut(a) Aut(g) to be induced from an automorphism in Auta(e) lies in H2(g, a)0 . Then we consider the family of Heisenberg Lie superalgebras and show that not all pairs are inducible in this family. We also give some necessary and sufficient conditions for inducibility of pairs arising in this family. ©2020 Heldermann Verlag.