On the possibility of group-theoretic description of an equivalence relation connected to the problem of covering subset in finite fields with cosets of linear subspaces

Let $F_q^n$ be an $n$-dimensional vector space over a finite field $F_q$. Let $C(F_q^n)$ denote the set of all cosets of linear subspaces in $F_q^n$. Cosets $H_1,H_2,\dots ,H_s$ are called exclusive if $H_i\not \subseteq H_j, 1\leq i<j \leq s$. A permutation $f$ of $C(F_q^n)$ is called a $C$-permutation if for any exclusive cosets $H,H_1,H_2,\dots ,H_s$ such that $H\subseteq H_1\cup H_2\cup \dots \cup H_s$, we have: i) cosets $f(H),f(H_1),f(H_2),\dots ,f(H_s)$ are exclusive; ii) cosets $f^{-1}(H),f^{-1}(H_1),f^{-1}(H_2),\dots ,f^{-1}(H_s)$ are exclusive; iii) $f(H)\subseteq f(H_1)\cup f(H_2)\cup \dots \cup f(H_s)$; iv) $f^{-1}(H)\subseteq f^{-1}(H_1)\cup f^{-1}(H_2)\cup \dots \cup f^{-1}(H_s)$. In this paper we show that the set of all $C$-permutations of $C(F_q^n)$ is the General Semiaffine Group of degree $n$ over $F_q$.