Properties of components for some classes of vectorial Boolean functions

In the class of invertible vectorial Boolean functions in $n$ variables with coordinate functions depending on all variables, we consider the subclasses $\mathcal{K}_{n}$ and $\mathcal{K}'_{n}$, where the functions are obtained using $n$ independent transpositions, respectively, from the identity permutation and from the permutation with coordinate functions essentially dependent on exactly one variable. We show that, for any $F=(f_1\ldots f_n)\in\mathcal{K}_{n}\cup\mathcal{K}'_{n}$ and $i\in\{1,\ldots,n\}$, the coordinate function $f_i$ has a single linear variable, each component function $vF$ with vector $v\in{\mathbb F}_2^n$ of a weight greater than $1$ has no fictitious and linear variables , the nonlinearity $N_{F}$, the degree $\deg F$, and the component algebraic immunity AI$_\text{comp}(F)$ are $2$, $n-1$, and $2$ respectively.