On Sylow tower of finite group with subnormal non-cyclic primary subgroups

Let $G$ be a finite group such that every non-cyclic maximal subgroups in its Sylow subgroups are subnormal in $G$. Suppose that a Sylow 2-subgroup of $G$ is either cyclic or self-normalizing. Under these assumptions, we prove that $G$ has a Sylow tower.