  The automorphism group $\operatorname{Aut}(F_n)$ of the free group $F_n$ acts
on a space $A_d(n)$ of Jacobi diagrams of degree $d$ on $n$ oriented arcs. We
study the $\operatorname{Aut}(F_n)$-module structure of $A_d(n)$ by using two
actions on the associated graded vector space of $A_d(n)$: an action of the
general linear group $\operatorname{GL}(n,Z)$ and an action of the graded Lie
algebra $\mathrm{gr}(\operatorname{IA}(n))$ of the IA-automorphism group
$\operatorname{IA}(n)$ of $F_n$ associated with its lower central series. We
extend the action of $\mathrm{gr}(\operatorname{IA}(n))$ to an action of the
associated graded Lie algebra of the Andreadakis filtration of the endomorphism
monoid of $F_n$. By using this action, we study the
$\operatorname{Aut}(F_n)$-module structure of $A_d(n)$. We obtain an
indecomposable decomposition of $A_d(n)$ as $\operatorname{Aut}(F_n)$-modules
for $n\geq 2d$. Moreover, we obtain the radical filtration of $A_d(n)$ for
$n\geq 2d$ and the socle of $A_3(n)$.
