On one-to-one property of a vectorial Boolean function of the special type

$\mathrm{S}$-boxes are widely used in cryptography. In particular, they form important components of SP and Feistel networks. Mathematically, $\mathrm{S}$-box is a vectorial Boolean function $F:\mathbb{F}_{2}^{n} \to \mathbb{F}_{2}^{m}$ that should satisfy several cryptographic properties. Usually $n=m$. We study one-to-one property of a vectorial Boolean function constructed in a special way on the base of a Boolean function and a permutation on $n$ elements. The number of all one-to-one functions of this type is calculated.