$H$-percolation with a random $H$

In $H$-percolation, we start with an Erdős–Rényi graph ${\mathcal{G}}_{n,p}$ and then iteratively add edges that complete copies of $H$. The process percolates if all edges missing from ${\mathcal{G}}_{n,p}$ are eventually added. We find the critical threshold $p_c$ when $H={\mathcal{G}}_{k,1\mathrm{/}2}$ is uniformly random, solving a problem of Balogh, Bollobás and Morris.