  A subgroup $H$ of a group $G$ is said to be an $IC\Phi$-subgroup of $G$ if $H
\cap [H,G] \le \Phi(H)$. We analyze the structure of a finite group $G$ under
the assumption that some given subgroups of $G$ are $IC\Phi$-subgroups of $G$.
A new characterization of finite abelian groups and some new criteria for
$2$-nilpotence and nilpotence of finite groups will be obtained. Moreover, we
will obtain two criteria for a finite group to lie in a given solvably
saturated formation containing the class of finite supersolvable groups.
