The inducibility of oriented stars

We consider the problem of maximizing the number of induced copies of an oriented star $S_{k,\mathrm{\ell }}$ in digraphs of given size, where the center of the star has out-degree $k$ and in-degree $\mathrm{\ell }$. The case $k\mathrm{\ell }=0$ was solved by Huang. Here, we asymptotically solve it for all other oriented stars with at least seven vertices.