Subgroups of a group of order $mp^{\alpha}$

Question

I would like to solve the following problem,

Let $G$ be a finite group with order $mp^{\alpha}, (m,p)= 1$. How can I show that for each $|H| = p^n, 1\le n \le \alpha-1$, $H \leqslant G$, there exists $K \leqslant G, |K| = p^{n+1}$ such that $H \unlhd K$?

Now I know how to show this for a $p$-group, but how do I show for a group as above? I also have a doubt that is this even true, as I have a doubt that does all $p$-power groups exist for $G$ or not?

Any Hints? Thank you.

Answer 1

An important part of Sylow's theorem is that every $p$-subgroup is contained in a Sylow $p$-subgroup.

So there is some $P\le G$ with $|P|=p^\alpha$ and $H\le P$.

You can prove the result for $P$, so there is some $K\le P\le G$ with $|K|=p^{n+1}$ and $H\trianglelefteq K$.