You will need to get the child's birth cert to find it.
We might need to do a bit more digging to get the right birth - but IF the middle initial A stands for Alfred, then this looks like it
Birth
Jul/Aug/Sep 1872
West Derby
8b 480
George Alfred Hall
(oooh, I see groom is one step ahead, and has found the Lancsbmd entry, I had forgotten about that)