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

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