Isospectral nearly K\"ahler manifolds

We give a systematic way to construct almost conjugate pairs of finite subgroups of $Spin(2n+1)$ and $Pin(n)$ for $n\in \mathbb{N}$ sufficiently large. As a geometric application, we give an infinite family of pairs $M_1^{d_n}$ and $M_2^{d_n}$ of nearly K\"ahler manifolds that are isospectral for the Dirac and Laplace operator with increasing dimensions $d_n>6$. We provide additionally a computation of the volume of (locally) homogeneous six dimensional nearly K\"ahler manifolds and investigate the existence of Sunada pairs in this dimension.