  Let $\left( 1\to N_n\to G_n\to Q_n\to 1 \right)_{n\in \mathbb{N}}$ be a
sequence of extensions of finitely generated groups with uniformly finite
generating subsets. We show that if the sequence $\left( N_n \right)_{n\in
\mathbb{N}} $ with the induced metric from the word metrics of $\left( G_n
\right)_{n\in \mathbb{N}} $ has property A, and the sequence $\left( Q_n
\right)_{n\in \mathbb{N}} $ with the quotient metrics coarsely embeds into
Hilbert space, then the coarse Baum-Connes conjecture holds for the sequence
$\left( G_n \right)_{n\in \mathbb{N}}$, which may not admit a coarse embedding
into Hilbert space. It follows that the coarse Baum-Connes conjecture holds for
the relative expanders and group extensions exhibited by G. Arzhantseva and R.
Tessera, and 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 in particular solves an open problem raised by
G. Arzhantseva and R. Tessera \cite{Arzhantseva-Tessera 2015}.
