On the normal complement problem for finite group algebras of Abelian-by-cyclic groups

Assume $F$ is a finite field of order $p^f$ and $q$ is an odd prime for which $p^f-1=s{q}^m$, where $m\ge 1$ and $(s,q)=1$. In this article, we obtain the order of symmetric and unitary subgroup of the semisimple group algebra $F{C}_q\mathrm{.}$ Further, for the extension $G$ of $C_q=⟨b⟩$ by an abelian group $A$ of order $p^n$ with $C_A(b)=\{e\}$, we prove that if $m>1,$ or $(s+1)\ge q$ and $2n\ge f(q-1)$, then $G$ does not have a normal complement in $V(FG)$.