$C_5$ is almost a fractalizer

We determine the maximum number of induced copies of a 5-cycle in a graph on $n$ vertices for every $n$. Every extremal construction is a balanced iterated blow-up of the 5-cycle with the possible exception of the smallest level where for $n=8$, the M\"obius ladder achieves the same number of induced 5-cycles as the blow-up of a 5-cycle on 8 vertices. This result completes work of Balogh, Hu, Lidick\'y, and Pfender [Eur. J. Comb. 52 (2016)] who proved an asymptotic version of the result. Similarly to their result, we also use the flag algebra method but we extend its use to small graphs.