Problem: Let $p$ be a prime and $n$ such that $\gcd(n,p)=1$. Then there exists $k_0$ such that for all $k \geq k_0$ any group of order $p^kn$ is not simple.
Attempt: I tried using the $p$-Sylow. Let us define $n_p$ the number of $p$-Sylow. Then we have that $n_p $ divide $n$ and that $n_p=1+hp$ for some $h \geq 1$. The cardinality of elements of order $p^a$ is more than $p^{k-1}(p-1)n_p + p^{k-1} \geq p^{k+1}$ but this does not help.
Trivial remark: If $n$ is a prime and $n<p$ then the $p$-Sylow is normal.
Any help will be appreciated.