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The main object of study of this paper is the notion of 3-Lie superalgebraswith superderivations. We consider a representation $(\Phi,\mathcal{P})$ of a$3$-Lie superalgebra $\mathcal{Q}$ on $\mathcal{P}$ and construct first-ordercohomologies by using superderivations of $\mathcal{P},\mathcal{Q}$ whichinduces a Lie superalgebra $\mathcal{T}_{\Phi}$ and its representation $\Psi$.Then we consider abelian extensions of $3$-Lie superalgebras of the form$0\rightarrow \mathcal{P}\hookrightarrow \mathcal{L}\rightarrow\mathcal{Q}\rightarrow 0$ with $[\mathcal{P},\mathcal{P},\mathcal{L}]=0$ andconstruct an obstruction class to extensibility of a compatible pair ofsuperderivation. Moreover we prove that a pair of superderivation is extensibleif and only if its obstruction class is trivial under some suitable conditions.

Cohomology, superderivations, and abelian extensions of $3$-Lie superalgebras