The main object of the study of this paper is the notion of 3-Lie superalgebras with superderivations. We consider a representation (, P) of a 3-Lie superalgebra Q on P and construct first-order cohomologies by using superderivations of P and Q which induce a Lie superalgebra T_ and its representation. Then, we consider an abelian extension of 3-Lie superalgebras of the form 0 P L Q 0 and construct an obstruction class to the extensibility of a compatible pair of superderivations. Moreover, we prove that a pair of superderivations is extensible if and only if its obstruction class is trivial under some suitable conditions.
Nandi et al. (Wed,) studied this question.