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Let $(W,S)$ be a Coxeter system, let $G$ be a finite solvable group ofautomorphisms of $(W,S)$ and let $\varphi$ be a weight function which isinvariant 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-Lusztigcells associated to $(W,\varphi)$ and to $(W^G,\varphi_G)$, provided thatLusztig's Conjectures hold.

Automorphisms of Coxeter groups and Lusztig's conjectures for Hecke algebras with unequal parameters