Radio won't turn off with key removed
 

I just picked up this 2004 Alero with 51K miles. The car is literally like new except for this issue. Everything is stock original. Here's the scenario:

Turn key to on position or start engine
Radio plays and works perfectly
With radio "on", turn key to off position, remove key, open door
Radio still plays,
Get out of car and lock doors with remote, radio still is on.
unlock doors with remote, open and shut door(s), re-lock doors with remote,
radio then turns off.

Can someone please tell me what to check to solve this problem? This is my sons first car and I want to make sure these issues are fixed before I turn it over to him.

Thanks
ZM

https://cimg4.ibsrv.net/gimg/oldsmob...59c7b4d737.jpg

Anyone???? ideas????

A new door lock actuator and It's fixed!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

https://cimg3.ibsrv.net/gimg/oldsmob...b3aa6cf5f9.png