Inspired by the Basilica group \mathcal B , we describe a general construction which allows us to associate to any group of automorphisms G \leq \operatorname{Aut}(T) of a rooted tree T a family of Basilica groups \operatorname{Bas}_s(G) , s \in \mathbb{N}_+ . For the dyadic odometer \mathcal O_2 , one has \mathcal B = \operatorname{Bas}_2(\mathcal O_2) . We study which properties of groups acting on rooted trees are preserved under this operation. Introducing some techniques for handling \operatorname{Bas}_s(G) , in case G fulfills some branching conditions, we are able to calculate the Hausdorff dimension of the Basilica groups associated to certain \mathsf{GGS} -groups and of generalisations of the odometer, \mathcal O_m^d . Furthermore, we study the structure of groups of type \operatorname{Bas}_s(\mathcal O_m^d) and prove an analogue of the congruence subgroup property in the case m = p , a prime.