Two Inverse results

Let $A$ be a subset of group $G_0$ with $\mathrm{\mid }{A}^{-1}A\mathrm{\mid }\le 2\mathrm{\mid }A\mathrm{\mid }-2.$ We show that there are an element $a\in A$ and a non-null proper subgroup $H$ of $G$ such that one of the following holds: \begin{itemize} \item $x^{-1}Hy\subset A^{-1}A,$ for all $(x,y)\in A^2\setminus (Ha{)}^2,$ \item $xH{y}^{-1}\subset A{A}^{-1},$ for all $(x,y)\in A^2\setminus (aH{)}^2\mathrm{.}$ \end{itemize} where $G$ is the subgroup generated by $A^{-1}A\mathrm{.}$ Assuming that $A^{-1}A\mathrm{\ne }G$ and that $\mathrm{\mid }{A}^{-1}A\mathrm{\mid }<\frac{5\mathrm{\mid }A\mathrm{\mid }}{3},$ we show that there are a normal subgroup $K$ of $G$ and a subgroup $H$ with $K\subset H\subset A^{-1}A $and$2|K|\ge |H|$suchthat$$A^{-1}AK=KA^{-1}A=A^{-1}A
$\text{and}$\ 6|K|\ge |A^{-1}A|=3|H|.$$