About the components of some classes of invertible 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 $\mathcal{K}'_{n}$, the functions in which are obtained using $n$ independent transpositions, respectively, from the identity permutation and from the permutation, each coordinate function of which essentially depends on some one variable. It is shown that, for any $F=(f_1\ldots f_n)\in\mathcal{K}_{n}\cup\mathcal{K}'_{n}$ and $i=1,\ldots,n$, the coordinate function $f_i$ has a single linear variable, the component function $vF$ has no nonessential and linear variables for each vector $v\in{\mathbb F}_2^n$ weight of which is greater than $1$, the nonlinearity, the degree, and the component algebraic immunity are $2$, $n-1$, and $2$ respectively.