Is $H$ a subgroup of $\Gamma$?
The constant mapping $h_2$ equal to $2$ belongs to $H$. Its inverse (namely the constant mapping equal to $1/2$) doesn't belong to $H$ though. So $H$ isn't a subgroup of $\Gamma$.
The constant mapping $h_2$ equal to $2$ belongs to $H$. Its inverse (namely the constant mapping equal to $1/2$) doesn't belong to $H$ though. So $H$ isn't a subgroup of $\Gamma$.