$\mathrm{S}$-blocks with maximum component algebraic immunity on a small number of variables

Let $\pi$ be a permutation on $ n $ elements, $f$ be a Boolean function in $n$ variables. Define a vector Boolean function $F_\pi:\mathbb{F}_2^n\rightarrow\mathbb{F}_2^n$ as $F_\pi(x) = (f(x), f(\pi(x)), \cdots, f (\pi^{n-1}(x))))$. In this paper, we study the component algebraic immunity of the vector Boolean function $F_\pi$ as a function of the Boolean function $f$ and the permutation $\pi$ for $n = 3, 4, 5$. We obtain complete sets of Boolean and, partly, vector Boolean functions with maximum algebraic immunity in $3, 4$ and $5$ variables. If the function $F_\pi$ has maximum algebraic immunity, then the permutation $\pi$ is full cycle.