  Let $G$ be a finite group and $H$ be a subgroup of $G$. Then $H$ is said to
be a $p$-$CAP$-subgroup of $G$, if $H$ covers or avoids any $pd$-chief factor
of $G$. Furthermore, $H$ is said to be a strong $p$-$CAP$-subgroup of $G$, if
for any $H \leq K \leq G$, $H$ is a $p$-$CAP$-subgroup of $K$. A subgroup $L$
is called an $ICSPC$-subgroup of $G$, if $[L,G] \cap L \leq L_{spcG}$, where
$L_{spcG}$ denotes a strong $p$-$CAP$-subgroup of $G$ contained in $L$. In this
paper, we investigate the structure of $G$ under the assumption that certain
subgroups of $G$ are $ICSPC$-subgroups of $G$. Characterizations for
$p$-nilpotency and solvably saturated formation are obtained. We also get
several criteria for the structure of $G$ under the assumption that certain
subgroups of $G$ are $ICSPC$-subgroups of $G$.
