Let $n$, $m\in \mathbb{N}$. Let $p$, $q$ be primes with $q^{n}|p-1$. Let $H$ the semidirect product of a cyclic group $A=C_{p^{\large{m}}}$ by a cyclic group $B=C_{q^{\large{n+1}}}$ which induces an automorphism of order $q^{n}$ on $A$ (i.e. I mean that if $B=<b>$ and $\alpha \in Aut(A)$ with $|\alpha|=q^{n}$, then $b$ acts on $A$ as $\alpha$ )
Has $H$ the structure of a Frobenius complement?