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In this paper, we define the induced modules of Lie algebra ad$(B)$associated with a 3-Lie algebra $B$-module, and study the relation between3-Lie algebra $A_{\omega}^{\delta}$-modules and induced modules of innerderivation algebra ad$(A_{\omega}^{\delta})$. We construct two infinitedimensional intermediate series modules of 3-Lie algebra $A_{\omega}^{\delta}$,and two infinite dimensional modules $(V, \psi_{\lambda\mu})$ and $(V,\phi_{\mu})$ of the Lie algebra  ad$(A_{\omega}^{\delta})$, and prove that only $(V, \psi_{\lambda0})$ and$(V, \psi_{\lambda1})$ are induced modules.