In this talk, we will discuss a class of substitutions whose associated subshifts are invariant under the action of some finite abelian group $G$. The geometry of these substitutions derives from digit tilings and they are allowed to have disconnected supertiles. $G$-invariance allows one to show that the associated dynamical system is a compact group extension via a suitable skew product, which has concrete and verifiable consequences to the underlying function space and the Lebesgue spectral types allowable for spectral measures. Illustrative examples will be provided throughout.

This is joint work with Natalie Frank.