  Let $G$ be a finite group and let $h$ be a positive integer. A
$\text{BH}(G,h)$ matrix is a $G$-invariant $|G|\times |G|$ matrix $H$ whose
entries are complex $h$th roots of unity such that $HH^*=|G|I_{|G|}$, where
$H^*$ denotes the complex conjugate transpose of $H$, and $I_{|G|}$ is the
identity matrix of order $|G|$. In this paper, we give three new constructions
of $\text{BH}(G,h)$ matrices. The first construction is the first known family
of $\text{BH}(G,h)$ matrices in which $G$ does not need to be abelian. The
second and the third constructions are two families of $\text{BH}(G,h)$
matrices in which $G$ is a finite local ring.
