Cryptographic properties of a simple S-box construction based on a Boolean function and a permutation

We propose a simple method of constructing S-boxes using Boolean functions and permutations. Let n be an arbitrary permutation on n elements, f be a Boolean function in n variables. Define a vectorial Boolean function Fn : F^ F^ as Fn(x) = = (f (x), f (n(x)), f (n2(x)),..., f (nn-1(x))). We study cryptographic properties of Fn such as high nonlinearity, balancedness, low differential 5-uniformity in dependence on properties of f and n for small n