  We generalize the combinatorial approaches of Rapaport and Higgins--Lyndon to
the Whitehead algorithm. We show that for every automorphism $\varphi$ of a
free group $F$ and every word $u\in F$ there exists a finite multiset of words
$S_{u,\varphi}$ satisfying the following property: For every cyclic word $w$,
the number of times $u$ appears as a subword of $\varphi\left(w\right)$ depends
only on the appearances of words in $S_{u,\varphi}$ as subwords of $w$. We use
this fact to construct a faithful representation of
$\text{Out}\left(F_{n}\right)$ on an inverse limit of $\mathbb{Z}$-modules, so
that each automorphism is represented by sequence of finite rectangular
matrices, which can be seen as successively better approximations of the
automorphism.
