HELP PLEASE: 99 Nissan Maxima Battery Dying???

Ok, here is the on going problem with my sisters car, the battery keeps being drained, so my father went out and bought a new alternator, everything seemed fine, the car ran, shut it off, started it again and and it was fine. now we let it sit for about 4 days went to start it and its deader than a door nail… WTF you might say well i say the same thing… it makes no sense, we tested the battery and it reads 12.4v, tested the alt. it reads 12.2, however when you turn the car on the alternator reads 4.5v

there is something wrong, does anyone have any suggestions that i might check or something i should be looking for???

any help guys would greatly be apperciated!