Higher index theory for spaces with an FCE-by-FCE structure

Let $(1\to N_n\to G_n\to Q_n\to 1{)}_{n\in \mathbb{N}}$ be a sequence of extensions of finite groups. Assume that the coarse disjoint unions of $(N_n{)}_{n\in \mathbb{N}}$, $(G_n{)}_{n\in \mathbb{N}}$ and $(Q_n{)}_{n\in \mathbb{N}}$ have bounded geometry. The sequence $(G_n{)}_{n\in \mathbb{N}}$ is said to have an \emph{FCE-by-FCE structure}, if the sequence $(N_n{)}_{n\in \mathbb{N}}$ and the sequence $(Q_n{)}_{n\in \mathbb{N}}$ admit \emph{a fibred coarse embedding} into Hilbert space. In this paper, we show that the coarse Novikov conjecture holds for spaces with an FCE-by-FCE structure.