Construction of a substitution on $\mathbb{F}_2^n$ based on a single Boolean function

The following construction of a vector Boolean function is considered: $F(x)=\big(f(x),f(\pi(x)),f(\pi^2(x)),\ldots, f(\pi^{n-1}(x))\big)$, where $\pi\in\mathbb{S}_n$, $f$ is a $n$-dimensional Boolean function. Some necessary conditions for $F$ to be a bijection are proved, namely: $f$ must be balanced, $f(0^n)\neq f(1^n)$, $\pi$ must be full cycle substitution, $f\neq\mathrm{const}$ on any cycle of substitution $\pi'$, where $\pi'(a_1\ldots a_n)=(a_{\pi(1)}\ldots a_{\pi(n)})$ for all $a_1\ldots a_n\in\mathbb{F}_2^n$.