Fix $k\ge 2$ and let $H$ be a graph with $\chi (H)=k+1$ containing a critical edge. We show that for sufficiently large $n,$ the unique $n$-vertex $H$-free graph containing the maximum number of cycles is $Tk(n)$. This resolves both a question and a conjecture of Arman, Gunderson and Tsaturian \cite{Gund1}.