Cardinalities of definable sets in superstructures over models

We introduce the notion of a superstructure over a model. This is a generalization of the notion of the hereditarily finite superstructure $\mathbb{HF}(\mathfrak M)$ over a model $\mathfrak M$. We consider the question on cardinalities of definable (interpretable) sets in superstructures over $\lambda$-homogeneous and $\lambda$-saturated models.