  Let $\left( 1\to N_m\to G_m\to Q_m\to 1 \right)_{m\in \mathbb{N}}$ be a
sequence of extensions of finite groups such that their coarse disjoint unions
have bounded geometry. In this paper, we show that if the coarse disjoint
unions of $\left( N_m \right)_{m\in \mathbb{N}} $ and $\left( Q_m \right)_{m\in
\mathbb{N}} $ are coarsely embeddable into Hilbert space, then the coarse
Baum-Connes conjecture holds for the coarse disjoint union of $\left( G_m
\right)_{m\in \mathbb{N}}$.
  As an application, the coarse Baum-Connes conjecture holds for the relative
expanders constructed by G. Arzhantseva and R. Tessera, and the special box
spaces of free groups discovered by T. Delabie and A. Khukhro, which do not
coarsely embed into Hilbert space, yet do not contain a weakly embedded
expander. This enlarges the class of metric spaces known to satisfy the coarse
Baum-Connes conjecture. In particular, it solves an open problem raised by G.
Arzhantseva and R. Tessera on the coarse Baum-Connes conjecture for relative
expanders.
