  We show that if H is a cocommutative Hopf algebra, then there is a natural
action of Aut(F_n) on the nth tensor power of H which induces an Out(F_n)
action on a quotient \overline{H^{\otimes 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 [J. Conant, The Johnson cokernel and the Enomoto-Satoh
invariant, Algebraic and Geometric Topology, 15 (2015), no. 2, 801--821.]
projects to take values in the top dimensional cohomology of Out(F_n) with
coefficients in \overline{H^{\otimes n}}. We analyze the n=2 case, getting
large families of obstructions generalizing the abelianization obstructions of
[J. Conant, M. Kassabov, K. Vogtmann, Higher hairy graph homology, Journal of
Topology, Geom. Dedicata 176 (2015), 345--374.].
