That sounds about right, but the Beatles had never set foot in the 2I's or even London by that point.
I think Williams was telegrammed from Germany about them needing a new band and he picked the Beatles as he was the manager at that point.
What George Harrison said in an interview is that Larry Parnes had persuaded either Derry and the Seniors to give up their jobs and let him manage them which they did but they never got any work so they decided to travel to London with Allan Williams and find Larry Parnes and beat him up. Williams told them to take their instruments as they might get a gig. They were jamming in the 2I's when they were noticed by Bruno Koschmider from Hamburg who was looking for a group to play at the Kaiserkeller, he picked them up and away they went.
Later on he sent for another band and that was The Beatles and the rest they say, is history.
So sorry Hank, but you've been getting mixed up
