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This article explores L∞ structures on 3-dimensional vector spaces with both Zopf- and Zopf2-gradings. Since Zopf-graded L∞ algebras are special cases of Zopf2-graded algebras in the induced Zopf2-grading, there are generally fewer Zopf-graded L∞ structures on a given space. However, degree zero automorphisms (rather than even automorphisms) determine equivalence in a Zopf-graded space. We therefore find nontrivial examples in which the map from the Zopf-graded moduli space to the Zopf2-graded moduli space is bijective, injective but not surjective, or surjective but not injective. Additionally, we study how the codifferentials in the moduli spaces deform into other nonequivalent codifferentials.
