We show that if H is a cocommutative Hopf algebra, then there is a natural action of Aut(F n ) on H ⊗n which induces an Out(F n ) action on a quotient H ⊗n . In the case when H = T(V) is the tensor algebra, we show that the invariant Tr C of the cokernel of the Johnson homomorphism studied in [5] projects to take values in H vcd (Out(F n ); H ⊗n ). We analyze the n = 2 case, getting large families of obstructions generalizing the abelianization obstructions of [8].arXiv:1509.03236v1 [math.AT] 10 Sep 2015where GL 2 (Z) acts on Sym(L (2) ) ⊗2 ∼ = Sym(L (2) ⊗ k 2 ) via the standard action on k 2 , which induces the action of GL 2 (Z) on Sym(L (2) ) ⊗2 . This latter group can be computed via the methods of [7], leading to many families of obstructions [λ] Sp ⊗ M k and [λ] Sp ⊗ S k , where M k , S k denote spaces of modular (respectively cusp) forms of weight k. The simplest new families of obstructions one gets are [2k − 1, 1 2 ] Sp ⊗ S 2k+2 ⊂ C 2k+5 , and ([2k + 1, 1 2 ] Sp ⊕ [2k, 2, 1] Sp ⊕ [2k, 1 3 ] Sp ) ⊗ M 2k+2 ⊂ C 2k+7 .