On preserving the structure of a subspace by a vectorial Boolean function

We consider the following property of a function $F: \mathbb{F}_2^{n} \to \mathbb{F}_2^{m}$: $F$ preserves the structure of an affine subspace $U \subseteq \mathbb{F}_2^{n}$ if $F(U) = \{F(x) : x \in U\}$ is an affine subspace of $\mathbb{F}_2^{m}$. The connection between this property and the existence of component functions of $F$ whose restriction to the subspace is constant is established. Estimations for the nonlinearity and the order of differential uniformity of such $F$ are provided. We also prove that the set of dimensions of affine subspaces whose structure is preserved by the multiplicative inversion function is the smallest among all one-to-one monomial functions.