Verbal width in anabelian groups

The class $A$ of anabelian groups is defined as the collection of finite
groups without abelian composition factors. We prove that the commutator word
$[x_1,x_2]$ and the power word $x_1^p$ have bounded width in $A$ when $p$ is an
odd integer. By contrast the word $x^{30}$ does not have bounded width in $A$.
On the other hand any given word $w$ has bounded width for those groups in $A$
whose composition factors are sufficiently large as a function of $w$. In the
course of the proof we establish that sufficiently large almost simple groups
cannot satisfy $w$ as a coset identity.