  Let $N$ be a normal subgroup of a finite group $G$. For a faithful $N$-set
$\Delta$, applying the university embedding theorem one can construct a
faithful $G$-set $\Omega$. In this short note, it is proved that if the
$2$-closure of $N$ in $\Omega$ is equal to $N$, then the $2$-closure of $N$ in
$\Delta$ is also equal to $N$; in addition, it is proved that any abelian
normal subgroup of a finite totally $2$-closed group is cyclic; finally, it is
proved that if a finite nilpotent group is a direct of two nilpotent subgroups
where the two factors have coprime orders and both of them are totally 2-closed
then G is totally $2$-closed. As corollaries, several well-known results on
finite totally 2-closed groups are reproved in more simple ways.
