  We consider an action of the automorphism group $\mathrm{Aut}(F_n)$ of the
free group $F_n$ of rank $n$ on the filtered vector space $A_d(n)$ of Jacobi
diagrams of degree $d$ on $n$ oriented arcs. This action induces on the
associated graded vector space of $A_d(n)$, which is identified with the space
$B_d(n)$ of open Jacobi diagrams, an action of the general linear group
$\mathrm{GL}(n,Z)$ and an action of the graded Lie algebra of the
IA-automorphism group of $F_n$ associated with its lower central series. We use
these actions on $B_d(n)$ to study the $\mathrm{Aut}(F_n)$-module structure of
$A_d(n)$. In particular, we consider the case where $d=2$ in detail and give an
indecomposable decomposition of $A_2(n)$. We also construct a polynomial
functor $A_d$ of degree $2d$ from the opposite category of the category of
finitely generated free groups to the category of filtered vector spaces, which
includes the $\mathrm{Aut}(F_n)$-module structure of $A_d(n)$ for all $n\geq
0$.
