On properties of the set of values of an arbitrary vector Boolean function

For an arbitrary vector Boolean function $F\colon\mathbb F_2^n\to\mathbb F_2^n$, some sufficient conditions that $\{F(x)\oplus F(x\oplus a)\colon x,a\in\mathbb F_2^n\}=\mathbb F_2^n$ are stated. This result is applied to researching metrical properties of APN functions.