Let $G$ be a group and $H$ be a simple non-abelian subgroup of $G$ which is ascendant in $G$. Is it true that $H$ is also subnormal in $G$?
Definition Let $G$ be a group and $H$ be a subgroup of $G$. Then we say that $H$ is ascendant in $G$ if we can find an ascendant (also of infinite length) normal series (not necessarily an invariant one) from $H$ to $G$.
Not true: take $H=A_5 \lt A_6 \lt G=A_7$. $A_5$ is not subnormal in $A_7$, since $A_7$ is simple.