  Let $(W,S)$ be a Coxeter system, let $G$ be a finite solvable group of
automorphisms of $(W,S)$ and let $\varphi$ be a weight function which is
invariant under $G$. Let $\varphi_G$ denote the weight function on $W^G$
obtained by restriction from $\varphi$. The aim of this paper is to compare the
${\mathbf{a}}$-function, the set of Duflo involutions and the Kazhdan-Lusztig
cells associated to $(W,\varphi)$ and to $(W^G,\varphi_G)$, provided that
Lusztig's Conjectures hold.
