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

For a graph $H$ denote by $\mathrm{emb}(H,m)$ the maximal number of labeled embeddings of $H$ in a graph of size $m$. Erdős posed the question of finding $\mathrm{emb}(H,m)$ for different values of $H$ and $m$. Following related asymptotic and stability results, we determine the value of $\mathrm{emb}(H,\left(\genfrac{}{}{0px}{}{n}{2}\right))$ for every graph $H$ with fractional independence number $v_H\mathrm{/}2$ 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.