Maximizing copies of a fixed graph in graphs with a prescribed number of edges

For a graph $H$ denote by $\operatorname{emb}\left(H, m\right)$ the maximal number of labeled embeddings of $H$ in a graph of size $m$. Erd\H{o}s posed the question of finding $\operatorname{emb}\left(H, m\right)$ for different values of $H$ and $m$. Following related asymptotic and stability results, we determine the value of $\operatorname{emb}\left(H, {\binom{n}{2}}\right)$ for every graph $H$ with fractional independence number $v_H/2$ and all sufficiently large $n$. We also show that for those $H$ (except for matchings) and $n$ the only graph with a maximal number of embeddings is $K_n$. This fully characterizes the graphs for which $K_n$ achieves the maximal number of embeddings.