On the Sylow 2-subgroup of a finite group

Question

Let $G$ be a finite group and $P$ be a Sylow 2-subgroup of $G$.

If at least one of maximal subgroups of $P$ is normal in $G$, then is $P$ cyclic?

Answer 1

An easy counterexample . . .

Let $G=Z_2\times Z_2$.

Then $P=G$, and $P$ is not cyclic.

Since $G$ is abelian, every subgroup of $P$ is normal in $G$.