It looks like the natural way to define this isomorphism is $(g_1, ..., g_k )\mapsto ([g_1], g_2, ..., g_k ) $ where $[g_1]$ is the congruence class of $g_1$ in $G/H$. I can see why this is onto, but I don't think $\Delta$ would be part of the kernel. So, it looks like I'm considering the wrong map, but what else is there to consider?
Thanks! I ran across this in a research paper, and it's just not coming to me why this is true.